Toptailieu.vn biên soạn và giới thiệu Phương pháp giải Cấp số nhân (HAY NHẤT 2024) gồm đầy đủ các phần: Lý thuyết, phương pháp giải, bài tập minh họa có lời giải chi tiết giúp học sinh làm tốt bài tập Toán 11 từ đó học tốt môn Toán. Mời các bạn đón xem:
Phương pháp giải Cấp số nhân (HAY NHẤT 2024)
1. Lý thuyết
a) Định nghĩa:
Cấp số nhân là một dãy số (hữu hạn hoặc vô hạn), trong đó kể từ số hạng thứ hai, mỗi số hạng đều là tích của số hạng đứng ngay trước nó với một số không đổi q.
Số q được gọi là công bội của cấp số nhân.
Nếu (un) là cấp số nhân với công bội q, ta có công thức truy hồi: un = un-1 . q với format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2218%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math134d8ebdaae4722206ee9c7d1c8%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2218.5%22%20y%3D%2218%22%3E%26%23x2208%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22math134d8ebdaae4722206ee9c7d1c8%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2232.5%22%20y%3D%2218%22%3E%26%23x2115%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2243.5%22%20y%3D%2211%22%3E*%3C%2Ftext%3E%3Ctext%20font-family%3D%22math134d8ebdaae4722206ee9c7d1c8%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2249.5%22%20y%3D%2218%22%3E.%3C%2Ftext%3E%3C%2Fsvg%3E)
Đặc biệt:
- Khi q = 0, cấp số nhân có dạng u1; 0; 0; … 0; …
- Khi q = 1, cấp số nhân có dạng u1; u1; … u1;…
- Khi u1 = 0 thì với mọi q, cấp số nhân có dạng 0; 0; 0; … 0; …
b) Số hạng tổng quát của cấp số nhân (un) được xác định bởi công thức:
un = u1 . qn - 1 với format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2216%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math12be2aae5d59f438a7e18a75d41%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2218.5%22%20y%3D%2216%22%3E%26%23x2265%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2230.5%22%20y%3D%2216%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math12be2aae5d59f438a7e18a75d41%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2236.5%22%20y%3D%2216%22%3E.%3C%2Ftext%3E%3C%2Fsvg%3E)
c) Tính chất
Ba số hạng uk - 1, uk, uk + 1 là ba số hạng liên tiếp của cấp số cộng khi và chỉ khi
với format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2216%22%3Ek%3C%2Ftext%3E%3Ctext%20font-family%3D%22math12be2aae5d59f438a7e18a75d41%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2217.5%22%20y%3D%2216%22%3E%26%23x2265%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2229.5%22%20y%3D%2216%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math12be2aae5d59f438a7e18a75d41%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2235.5%22%20y%3D%2216%22%3E.%3C%2Ftext%3E%3C%2Fsvg%3E)
(Hay
).
d) Tổng n số hạng đầu tiên của cấp số nhân được xác định bởi công thức:
format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A'round_brackets18549f92a457f2409'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMjwHLFQAAADMAAAATmNtYXDf7xCrAAABHAAAADxjdnQgBAkDLgAAAVgAAAASZ2x5ZmAOz2cAAAFsAAABJGhlYWQOKih8AAACkAAAADZoaGVhCvgVwgAAAsgAAAAkaG10eCA6AAIAAALsAAAADGxvY2EAAARLAAAC%2BAAAABBtYXhwBIgEWQAAAwgAAAAgbmFtZXHR30MAAAMoAAACOXBvc3QDogHPAAAFZAAAACBwcmVwupWEAAAABYQAAAAHAAAGcgGQAAUAAAgACAAAAAAACAAIAAAAAAAAAQIAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAo8AMGe%2F57AAAHPgGyAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACACgAKf%2F%2FAAAAKAAp%2F%2F%2F%2F2f%2FZAAEAAAAAAAAAAAFUAFYBAAAsAKgDgAAyAAcAAAACAAAAKgDVA1UAAwAHAAA1MxEjEyMRM9XVq4CAKgMr%2FQAC1QABAAD%2B0AIgBtAACQBNGAGwChCwA9SwAxCwAtSwChCwBdSwBRCwANSwAxCwBzywAhCwCDwAsAoQsAPUsAMQsAfUsAoQsAXUsAoQsADUsAMQsAI8sAcQsAg8MTAREAEzABEQASMAAZCQ%2FnABkJD%2BcALQ%2FZD%2BcAGQAnACcAGQ%2FnAAAQAA%2FtACIAbQAAkATRgBsAoQsAPUsAMQsALUsAoQsAXUsAUQsADUsAMQsAc8sAIQsAg8ALAKELAD1LADELAH1LAKELAF1LAKELAA1LADELACPLAHELAIPDEwARABIwAREAEzAAIg%2FnCQAZD%2BcJABkALQ%2FZD%2BcAGQAnACcAGQ%2FnAAAQAAAAEAAPW2NYFfDzz1AAMIAP%2F%2F%2F%2F%2FVre7u%2F%2F%2F%2F%2F9Wt7u4AAP7QA7cG0AAAAAoAAgABAAAAAAABAAAHPv5OAAAXcAAA%2F%2F4DtwABAAAAAAAAAAAAAAAAAAAAAwDVAAACIAAAAiAAAAAAAAAAAAAkAAAAowAAASQAAQAAAAMACgACAAAAAAACAIAEAAAAAAAEAABNAAAAAAAAABUBAgAAAAAAAAABAD4AAAAAAAAAAAACAA4APgAAAAAAAAADAFwATAAAAAAAAAAEAD4AqAAAAAAAAAAFABYA5gAAAAAAAAAGAB8A%2FAAAAAAAAAAIABwBGwABAAAAAAABAD4AAAABAAAAAAACAA4APgABAAAAAAADAFwATAABAAAAAAAEAD4AqAABAAAAAAAFABYA5gABAAAAAAAGAB8A%2FAABAAAAAAAIABwBGwADAAEECQABAD4AAAADAAEECQACAA4APgADAAEECQADAFwATAADAAEECQAEAD4AqAADAAEECQAFABYA5gADAAEECQAGAB8A%2FAADAAEECQAIABwBGwBSAG8AdQBuAGQAIABiAHIAYQBjAGsAZQB0AHMAIAB3AGkAdABoACAAYQBzAGMAZQBuAHQAIAAxADgANQA0AFIAZQBnAHUAbABhAHIATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlACAAUgBvAHUAbgBkACAAYgByAGEAYwBrAGUAdABzACAAdwBpAHQAaAAgAGEAcwBjAGUAbgB0ACAAMQA4ADUANABSAG8AdQBuAGQAIABiAHIAYQBjAGsAZQB0AHMAIAB3AGkAdABoACAAYQBzAGMAZQBuAHQAIAAxADgANQA0AFYAZQByAHMAaQBvAG4AIAAyAC4AMFJvdW5kX2JyYWNrZXRzX3dpdGhfYXNjZW50XzE4NTQATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlAAAAAAMAAAAAAAADnwHPAAAAAAAAAAAAAAAAAAAAAAAAAAC5B%2F8AAY2FAA%3D%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2234%22%3ES%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2215.5%22%20y%3D%2239%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math13ca958c2523eaed49890c46b08%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2228.5%22%20y%3D%2234%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2241.5%22%20y%3D%2234%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2250.5%22%20y%3D%2239%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math13ca958c2523eaed49890c46b08%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2261.5%22%20y%3D%2234%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2273.5%22%20y%3D%2234%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2282.5%22%20y%3D%2239%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math13ca958c2523eaed49890c46b08%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2293.5%22%20y%3D%2234%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22math13ca958c2523eaed49890c46b08%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22105.5%22%20y%3D%2234%22%3E...%3C%2Ftext%3E%3Ctext%20font-family%3D%22math13ca958c2523eaed49890c46b08%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22117.5%22%20y%3D%2234%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22129.5%22%20y%3D%2234%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22138.5%22%20y%3D%2239%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math13ca958c2523eaed49890c46b08%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22151.5%22%20y%3D%2234%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%22162.5%22%20x2%3D%22234.5%22%20y1%3D%2228.5%22%20y2%3D%2228.5%22%2F%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22168.5%22%20y%3D%2218%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22177.5%22%20y%3D%2223%22%3E1%3C%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Chú ý: Nếu q = 1 thì cấp số nhân là u1; u1; u1; … u1;.. khi đó Sn = n.u1.
2. Các dạng toán
Dạng 1. Xác định cấp số cộng và các yếu tố của cấp số nhân
Phương pháp giải:
- Dãy số (un) là một cấp số nhân khi và chỉ khi
không phụ thuộc vào n và q là công bội của cấp số nhân đó.
- Để xác định một cấp số nhân, ta cần xác định số hạng đầu và công bội. Ta thiết lập một hệ phương trình hai ẩn u1 và q. Tìm u1 và q.
- Tìm số hạng thứ n dựa vào công thức tổng quát: un = u1 . qn-1 hoặc công thức truy hồi un = un – 1 . q.
Ví dụ minh họa:
Ví dụ 1: Cho các dãy số sau, dãy số nào là cấp số nhân. Nếu là cấp số nhân hãy xác định số hạng đầu tiên và công bội:
a) 1; – 2; 4; – 8; 16; – 32; 64
b) Dãy (un): un = n.6n+1
c) Dãy (vn): vn = (– 1)n.32n.
Lời giải
a) Ta thấy format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%222.5%22%20x2%3D%2227.5%22%20y1%3D%2221.5%22%20y2%3D%2221.5%22%2F%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2210.5%22%20y%3D%2216%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2221.5%22%20y%3D%2216%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2215.5%22%20y%3D%2238%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2238.5%22%20y%3D%2227%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%2249.5%22%20x2%3D%2274.5%22%20y1%3D%2221.5%22%20y2%3D%2221.5%22%2F%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2262.5%22%20y%3D%2216%22%3E4%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2257.5%22%20y%3D%2239%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2268.5%22%20y%3D%2239%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2285.5%22%20y%3D%2227%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%2296.5%22%20x2%3D%22121.5%22%20y1%3D%2221.5%22%20y2%3D%2221.5%22%2F%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22104.5%22%20y%3D%2216%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22115.5%22%20y%3D%2216%22%3E8%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22109.5%22%20y%3D%2238%22%3E4%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22132.5%22%20y%3D%2227%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%22143.5%22%20x2%3D%22168.5%22%20y1%3D%2221.5%22%20y2%3D%2221.5%22%2F%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22156.5%22%20y%3D%2216%22%3E16%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22151.5%22%20y%3D%2239%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22162.5%22%20y%3D%2239%22%3E8%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22179.5%22%20y%3D%2227%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%22190.5%22%20x2%3D%22224.5%22%20y1%3D%2221.5%22%20y2%3D%2221.5%22%2F%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22198.5%22%20y%3D%2216%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22214.5%22%20y%3D%2216%22%3E32%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22208.5%22%20y%3D%2238%22%3E16%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22235.5%22%20y%3D%2227%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%22246.5%22%20x2%3D%22280.5%22%20y1%3D%2221.5%22%20y2%3D%2221.5%22%2F%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22264.5%22%20y%3D%2216%22%3E64%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22254.5%22%20y%3D%2239%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22270.5%22%20y%3D%2239%22%3E32%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22291.5%22%20y%3D%2227%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22307.5%22%20y%3D%2227%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22319.5%22%20y%3D%2227%22%3E2%3C%2Ftext%3E%3C%2Fsvg%3E)
Nên dãy số trên là cấp số nhân với số hạng đầu tiên là u1 = 1 và công bội q = – 2.
b) Ta có: un = n. 6n+1 thì un+1 = (n + 1).6n+2
Xét
phụ thuộc vào n
Nên dãy số trên không là cấp số nhân.
c) Ta có: vn = (– 1)n. 32n thì vn+1 = (– 1)n+1. 32(n+1)
Xét
không đổi.
Vậy dãy số trên là cấp số nhân với số hạng đầu tiên u1 = (– 1)1.32.1 = – 9 và công bội q = – 9.
Ví dụ 2: Cho cấp số nhân (un) thỏa mãn: format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A'brack_sm74a4ec520979a73b06ad972'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7PH4UAAADMAAAATmNtYXA3kjw6AAABHAAAAExjdnQgAQYDiAAAAWgAAAASZ2x5ZkyYQ7YAAAF8AAABLGhlYWQLyR8fAAACqAAAADZoaGVhAq0XCAAAAuAAAAAkaG10eDEjA%2FUAAAMEAAAAFGxvY2EAAEKZAAADGAAAABhtYXhwBJsEcQAAAzAAAAAgbmFtZW7QvZAAAANQAAAB5XBvc3QArQBVAAAFOAAAACBwcmVwu5WEAAAABVgAAAAHAAACDAGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg9AMD%2FP%2F8AAABVAABAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADgAAAAKAAgAAgACI6cjqCOpI6r%2F%2FwAAI6cjqCOpI6r%2F%2F9xa3FrcWtxaAAEAAAAAAAAAAAAAAAABVABUAQAAKwCMAIAAqAAHAAAAAgAAAAAA1QEBAAMABwAAMTMRIxcjNTPV1auAgAEB1qsAAQAAAAABAAFVAAcAIxgBsQAAP7EHAv2xAwf8sAQ8ALEAAD%2BwBzyxAwH2sQQE%2FTEwMTQ2NxUOARVrlWtAa8AqKivAQAAB%2F4AAAABVAVQACQAhGAGxBQA%2FsAE8sQYC%2FbAAPACxBQA%2FsAY8sQEBP7AAPDEwEyMUBxYVMzQnNlVVgIBVgIABVFVUVlVVVlQAAQAAAAABAAFUAAcAIxgBsQAAP7EHAv2xAwf8sAQ8ALEAAT%2BwBzyxAwH2sQQE%2FTEwERQWFzUuATVrlWtAAVRqwCorKcBAAAEAAAAAAFUBVAADACAYAbAEELAAPLADPLEBAvWwAjwAsAMvsAI8sQAB9bABPBEzESNVVQFU%2FqwAAQAAAAEAAIsesexfDzz1AAMEAP%2F%2F%2F%2F%2FVre5k%2F%2F%2F%2F%2F9Wt7mT%2FgP%2F%2FAdYBWAAAAAoAAgABAAAAAAABAAABVP%2F%2FAAAXcP%2BA%2F4AB1gABAAAAAAAAAAAAAAAAAAAABQDVAAABAAAAAQD%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%2FAAKNhQA%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%226%22%3E%26%23x23A7%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2211%22%3E%26%23x23AA%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2216%22%3E%26%23x23AA%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2221%22%3E%26%23x23AA%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2226%22%3E%26%23x23AA%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2231%22%3E%26%23x23A8%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2236%22%3E%26%23x23AA%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2241%22%3E%26%23x23AA%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2246%22%3E%26%23x23AA%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2251%22%3E%26%23x23AA%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm74a4ec520979a73b06ad972%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%223.5%22%20y%3D%2256%22%3E%26%23x23A9%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2216.5%22%20y%3D%2218%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2225.5%22%20y%3D%2223%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math12ed72e0d2d50af08c235c494fe%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2236.5%22%20y%3D%2218%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2248.5%22%20y%3D%2218%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2257.5%22%20y%3D%2223%22%3E5%3C%2Ftext%3E%3Ctext%20font-family%3D%22math12ed72e0d2d50af08c235c494fe%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2269.5%22%20y%3D%2218%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2287.5%22%20y%3D%2218%22%3E51%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2216.5%22%20y%3D%2247%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2225.5%22%20y%3D%2252%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math12ed72e0d2d50af08c235c494fe%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2236.5%22%20y%3D%2247%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2248.5%22%20y%3D%2247%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2257.5%22%20y%3D%2252%22%3E6%3C%2Ftext%3E%3Ctext%20font-family%3D%22math12ed72e0d2d50af08c235c494fe%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2269.5%22%20y%3D%2247%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2291.5%22%20y%3D%2247%22%3E102%3C%2Ftext%3E%3C%2Fsvg%3E)
a) Xác định công bội và hạng đầu tiên của cấp số nhân trên.
b) Xác định công thức tổng quát của cấp số nhân trên.
c) Tìm số hạng thứ 15 của cấp số cộng trên.
d) Số 12288 là số hạng thứ bao nhiêu của cấp số nhân.
Lời giải
a) Gọi q là công bội của cấp số nhân đã cho. Theo đề bài, ta có
 alt="open curly brackets table attributes columnalign left end attributes row cell u subscript 1 plus u subscript 5 equals 51 end cell row cell u subscript 2 plus u subscript 6 equals 102 end cell end table close left right double arrow open curly brackets table attributes columnalign left end attributes row cell u subscript 1 plus u subscript 1 q to the power of 4 equals 51 end cell row cell u subscript 1 q plus u subscript 1 q to the power of 5 equals 102 end cell end table close
left right double arrow open curly brackets table attributes columnalign left end attributes row cell u subscript 1 open parentheses 1 plus q to the power of 4 close parentheses equals 51 end cell row cell u subscript 1 q open parentheses 1 plus q to the power of 4 close parentheses equals 102 end cell end table 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Lấy hai vế của phương trình dưới chia cho hai vế của phương trình trên ta được q = 2.
Suy ra format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2226%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2213.5%22%20y%3D%2231%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1564b4c0e54101ac57a0cb68c16%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2225.5%22%20y%3D%2226%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%2236.5%22%20x2%3D%2280.5%22%20y1%3D%2220.5%22%20y2%3D%2220.5%22%2F%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2259.5%22%20y%3D%2215%22%3E51%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2242.5%22%20y%3D%2240%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1564b4c0e54101ac57a0cb68c16%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2254.5%22%20y%3D%2240%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2266.5%22%20y%3D%2240%22%3Eq%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2275.5%22%20y%3D%2233%22%3E4%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1564b4c0e54101ac57a0cb68c16%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2291.5%22%20y%3D%2226%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%22102.5%22%20x2%3D%22145.5%22%20y1%3D%2220.5%22%20y2%3D%2220.5%22%2F%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22124.5%22%20y%3D%2215%22%3E51%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22108.5%22%20y%3D%2240%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1564b4c0e54101ac57a0cb68c16%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22120.5%22%20y%3D%2240%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22132.5%22%20y%3D%2240%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22140.5%22%20y%3D%2233%22%3E4%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1564b4c0e54101ac57a0cb68c16%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22156.5%22%20y%3D%2226%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22169.5%22%20y%3D%2226%22%3E3%3C%2Ftext%3E%3C%2Fsvg%3E)
Vậy cấp số nhân có số hạng đầu tiên là u1 = 3 và công bội q = 2.
b) Số hạng tổng quát của cấp số nhân là un = u1. qn–1 nên un = 3.2n–1.
c) Số hạng thứ 15 của cấp số nhân là: u15 = 3.214 = 49152.
d) Giả sử số 12288 là số hạng thứ n của cấp số nhân, ta có:
.
Vậy số 12288 là số hạng thứ 13 của cấp số nhân.
Dạng 2. Tìm điều kiện để dãy số lập thành cấp số nhân. Chứng minh cấp số nhân
Phương pháp giải:
Sử dụng tính chất: Ba số hạng uk – 1 ; uk ; uk + 1 là ba số hạng liên tiếp của cấp số nhân khi và chỉ khi
.
Ví dụ minh họa:
Ví dụ 1: Tìm x sao cho các số 1; x2; 6 – x2 lập thành cấp số nhân.
Lời giải
Ta có: 1; x2; 6 – x2 lập thành cấp số nhân
 alt="left right double arrow x to the power of 4 equals 1. open parentheses 6 minus x squared close parentheses left right double arrow open square brackets table attributes columnalign left end attributes row cell x squared equals 2 end cell row cell x squared equals negative 3 text end text left parenthesis L o ạ i right parenthesis end cell end table close
left right double arrow x equals plus-or-minus square root of 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Vậy
thì các số trên lập thành cấp số nhân.
Ví dụ 2: Các số 5x – y; 2x + 3y; x + 2y lập thành cấp số cộng; các số (y + 1)2 ; xy + 1 ; (x – 1)2 lập thành cấp số nhân. Tìm x và y.
Lời giải

Vậy
.
Dạng 3. Tính tổng của một cấp số nhân
Phương pháp giải:
Tổng n số hạng đầu tiên Sn được xác định bởi công thức: format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A'round_brackets18549f92a457f2409'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMjwHLFQAAADMAAAATmNtYXDf7xCrAAABHAAAADxjdnQgBAkDLgAAAVgAAAASZ2x5ZmAOz2cAAAFsAAABJGhlYWQOKih8AAACkAAAADZoaGVhCvgVwgAAAsgAAAAkaG10eCA6AAIAAALsAAAADGxvY2EAAARLAAAC%2BAAAABBtYXhwBIgEWQAAAwgAAAAgbmFtZXHR30MAAAMoAAACOXBvc3QDogHPAAAFZAAAACBwcmVwupWEAAAABYQAAAAHAAAGcgGQAAUAAAgACAAAAAAACAAIAAAAAAAAAQIAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAo8AMGe%2F57AAAHPgGyAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACACgAKf%2F%2FAAAAKAAp%2F%2F%2F%2F2f%2FZAAEAAAAAAAAAAAFUAFYBAAAsAKgDgAAyAAcAAAACAAAAKgDVA1UAAwAHAAA1MxEjEyMRM9XVq4CAKgMr%2FQAC1QABAAD%2B0AIgBtAACQBNGAGwChCwA9SwAxCwAtSwChCwBdSwBRCwANSwAxCwBzywAhCwCDwAsAoQsAPUsAMQsAfUsAoQsAXUsAoQsADUsAMQsAI8sAcQsAg8MTAREAEzABEQASMAAZCQ%2FnABkJD%2BcALQ%2FZD%2BcAGQAnACcAGQ%2FnAAAQAA%2FtACIAbQAAkATRgBsAoQsAPUsAMQsALUsAoQsAXUsAUQsADUsAMQsAc8sAIQsAg8ALAKELAD1LADELAH1LAKELAF1LAKELAA1LADELACPLAHELAIPDEwARABIwAREAEzAAIg%2FnCQAZD%2BcJABkALQ%2FZD%2BcAGQAnACcAGQ%2FnAAAQAAAAEAAPW2NYFfDzz1AAMIAP%2F%2F%2F%2F%2FVre7u%2F%2F%2F%2F%2F9Wt7u4AAP7QA7cG0AAAAAoAAgABAAAAAAABAAAHPv5OAAAXcAAA%2F%2F4DtwABAAAAAAAAAAAAAAAAAAAAAwDVAAACIAAAAiAAAAAAAAAAAAAkAAAAowAAASQAAQAAAAMACgACAAAAAAACAIAEAAAAAAAEAABNAAAAAAAAABUBAgAAAAAAAAABAD4AAAAAAAAAAAACAA4APgAAAAAAAAADAFwATAAAAAAAAAAEAD4AqAAAAAAAAAAFABYA5gAAAAAAAAAGAB8A%2FAAAAAAAAAAIABwBGwABAAAAAAABAD4AAAABAAAAAAACAA4APgABAAAAAAADAFwATAABAAAAAAAEAD4AqAABAAAAAAAFABYA5gABAAAAAAAGAB8A%2FAABAAAAAAAIABwBGwADAAEECQABAD4AAAADAAEECQACAA4APgADAAEECQADAFwATAADAAEECQAEAD4AqAADAAEECQAFABYA5gADAAEECQAGAB8A%2FAADAAEECQAIABwBGwBSAG8AdQBuAGQAIABiAHIAYQBjAGsAZQB0AHMAIAB3AGkAdABoACAAYQBzAGMAZQBuAHQAIAAxADgANQA0AFIAZQBnAHUAbABhAHIATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlACAAUgBvAHUAbgBkACAAYgByAGEAYwBrAGUAdABzACAAdwBpAHQAaAAgAGEAcwBjAGUAbgB0ACAAMQA4ADUANABSAG8AdQBuAGQAIABiAHIAYQBjAGsAZQB0AHMAIAB3AGkAdABoACAAYQBzAGMAZQBuAHQAIAAxADgANQA0AFYAZQByAHMAaQBvAG4AIAAyAC4AMFJvdW5kX2JyYWNrZXRzX3dpdGhfYXNjZW50XzE4NTQATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlAAAAAAMAAAAAAAADnwHPAAAAAAAAAAAAAAAAAAAAAAAAAAC5B%2F8AAY2FAA%3D%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2234%22%3ES%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2215.5%22%20y%3D%2239%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19531528af376a9cba4b066060d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2228.5%22%20y%3D%2234%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%2239.5%22%20x2%3D%22111.5%22%20y1%3D%2228.5%22%20y2%3D%2228.5%22%2F%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2245.5%22%20y%3D%2218%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2254.5%22%20y%3D%2223%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22round_brackets18549f92a457f2409%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2261.5%22%20y%3D%2218%22%3E(%3C%2Ftext%3E%3Ctext%20font-family%3D%22round_brackets18549f92a457f2409%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22107.5%22%20y%3D%2218%22%3E)%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2267.5%22%20y%3D%2218%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19531528af376a9cba4b066060d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2279.5%22%20y%3D%2218%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2291.5%22%20y%3D%2218%22%3Eq%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22100.5%22%20y%3D%2211%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2263.5%22%20y%3D%2246%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19531528af376a9cba4b066060d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2275.5%22%20y%3D%2246%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2287.5%22%20y%3D%2246%22%3Eq%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19531528af376a9cba4b066060d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22116.5%22%20y%3D%2234%22%3E%2C%3C%2Ftext%3E%3Ctext%20font-family%3D%22round_brackets18549f92a457f2409%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22128.5%22%20y%3D%2234%22%3E(%3C%2Ftext%3E%3Ctext%20font-family%3D%22round_brackets18549f92a457f2409%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22168.5%22%20y%3D%2234%22%3E)%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22134.5%22%20y%3D%2234%22%3Eq%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19531528af376a9cba4b066060d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22148.5%22%20y%3D%2234%22%3E%26%23x2260%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22161.5%22%20y%3D%2234%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19531528af376a9cba4b066060d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22173.5%22%20y%3D%2234%22%3E.%3C%2Ftext%3E%3C%2Fsvg%3E)
Nếu q = 1 thì cấp số nhân là u1; u1; u1; … u1; … khi đó Sn = n.u1.
Ví dụ minh họa:
Ví dụ 1: Cho cấp số nhân (un)
a) (un) có số hạng tổng quát là: un = 2.( –3)k. Tính S15.
b) (un) có số hạng đầu là 18, số hạng thứ hai kia là 54, số hạng cuối bằng 39366. Tính tổng của tất cả các số hạng của cấp số nhân.
Lời giải
a) (un) có số hạng tổng quát là: un = 2. (– 3)k thì u1 = 2 và q = – 3
Tổng 15 số hạng đầu tiên của cấp số nhân là
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b) Số hạng đầu tiên u1 = 18
Số hạng thứ hai format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2216%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2213.5%22%20y%3D%2221%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1877ea5b692234ab93be555217a%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2225.5%22%20y%3D%2216%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2243.5%22%20y%3D%2216%22%3E54%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1877ea5b692234ab93be555217a%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2262.5%22%20y%3D%2216%22%3E%26%23x21D2%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2276.5%22%20y%3D%2216%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2285.5%22%20y%3D%2221%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2293.5%22%20y%3D%2216%22%3Eq%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1877ea5b692234ab93be555217a%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22107.5%22%20y%3D%2216%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22125.5%22%20y%3D%2216%22%3E54%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1877ea5b692234ab93be555217a%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22144.5%22%20y%3D%2216%22%3E%26%23x21D2%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22158.5%22%20y%3D%2216%22%3Eq%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1877ea5b692234ab93be555217a%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22172.5%22%20y%3D%2216%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22185.5%22%20y%3D%2216%22%3E3%3C%2Ftext%3E%3C%2Fsvg%3E)
Số hạng cuối format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2216%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2213.5%22%20y%3D%2221%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f39f8317fbdb1988ef4c628eb%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2226.5%22%20y%3D%2216%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2257.5%22%20y%3D%2216%22%3E39366%3C%2Ftext%3E%3C%2Fsvg%3E)
format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2212.5%22%20y%3D%2219%22%3E%26%23x21D4%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2228.5%22%20y%3D%2219%22%3Eu%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2237.5%22%20y%3D%2224%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2243.5%22%20y%3D%2219%22%3E.%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2250.5%22%20y%3D%2219%22%3Eq%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2259.5%22%20y%3D%2212%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2270.5%22%20y%3D%2212%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2279.5%22%20y%3D%2212%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2291.5%22%20y%3D%2219%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22122.5%22%20y%3D%2219%22%3E39366%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22157.5%22%20y%3D%2219%22%3E%26%23x21D4%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22178.5%22%20y%3D%2219%22%3E18%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22188.5%22%20y%3D%2219%22%3E.%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22194.5%22%20y%3D%2219%22%3E3%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22202.5%22%20y%3D%2212%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22213.5%22%20y%3D%2212%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22222.5%22%20y%3D%2212%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22234.5%22%20y%3D%2219%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22265.5%22%20y%3D%2219%22%3E39366%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2212.5%22%20y%3D%2251%22%3E%26%23x21D4%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2228.5%22%20y%3D%2251%22%3E3%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2236.5%22%20y%3D%2244%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2247.5%22%20y%3D%2244%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2256.5%22%20y%3D%2244%22%3E1%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2268.5%22%20y%3D%2251%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2281.5%22%20y%3D%2251%22%3E3%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2289.5%22%20y%3D%2244%22%3E7%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22105.5%22%20y%3D%2251%22%3E%26%23x21D4%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22121.5%22%20y%3D%2251%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f74227241c40435e2a58da8e5%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22135.5%22%20y%3D%2251%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22148.5%22%20y%3D%2251%22%3E8%3C%2Ftext%3E%3C%2Fsvg%3E)
Vậy  alt="text S end text subscript 8 equals fraction numerator u subscript 1 open parentheses 1 minus q to the power of n close parentheses over denominator 1 minus q end fraction equals fraction numerator 18. open parentheses 1 minus 3 to the power of 8 close parentheses over denominator 1 minus 3 end fraction equals 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Ví dụ 2: Tính tổng
a) format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A'math1b5dc6aeb84bcea29a76d1e9d00'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAAERjdnQgDVUNBwAAAWAAAAA6Z2x5ZoPi2VsAAAGcAAABcWhlYWQQC2qxAAADEAAAADZoaGVhCGsXSAAAA0gAAAAkaG10eE2rRkcAAANsAAAAEGxvY2EAHTwYAAADfAAAABRtYXhwBT0FPgAAA5AAAAAgbmFtZaBxlY4AAAOwAAABn3Bvc3QB9wD6AAAFUAAAACBwcmVwa1uragAABXAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADAAAAAIAAgAAgAAACsALgA9%2F%2F8AAAArAC4APf%2F%2F%2F9b%2F1P%2FGAAEAAAAAAAAAAAAAAVQDLACAAQAAVgAqAlgCHgEOASwCLABaAYACgACgANQAgAAAAAAAAAArAFUAgACrANUBAAErAAcAAAACAFUAAAMAA6sAAwAHAAAzESERJSERIVUCq%2F2rAgD%2BAAOr%2FFVVAwAAAQCAAFUC1QKrAAsASQEYsgwBARQTELEAA%2FaxAQT1sAo8sQMF9bAIPLEFBPWwBjyxDQPmALEAABMQsQEG5LEBARMQsAU8sQME5bELBfWwBzyxCQTlMTATIREzESEVIREjESGAAQBVAQD%2FAFX%2FAAGrAQD%2FAFb%2FAAEAAAEAIAAAAKAAgAADAC8YAbAEELAD1LADELAC1LADELAAPLACELABPACwBBCwA9SwAxCwAjywABCwATwwMTczFSMggICAgAACAIAA6wLVAhUAAwAHAGUYAbAIELAG1LAGELAF1LAIELAB1LABELAA1LAGELAHPLAFELAEPLABELACPLAAELADPACwCBCwBtSwBhCwB9SwBxCwAdSwARCwAtSwBhCwBTywBxCwBDywARCwADywAhCwAzwxMBMhNSEdASE1gAJV%2FasCVQHAVdVVVQAAAAABAAAAAQAA1XjOQV8PPPUAAwQA%2F%2F%2F%2F%2F9Y6E3P%2F%2F%2F%2F%2F1joTcwAA%2FyAEgAOrAAAACgACAAEAAAAAAAEAAAPo%2F2oAABdwAAD%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%2BgAAAAAAAAAAAAAAAAAAAAAAAAAAuQcRAACNhRgAsgAAABUUE7EAAT8%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3ES%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2215.5%22%20y%3D%2221%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2228.5%22%20y%3D%2216%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2241.5%22%20y%3D%2216%22%3E9%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2253.5%22%20y%3D%2216%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2270.5%22%20y%3D%2216%22%3E99%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2286.5%22%20y%3D%2216%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22107.5%22%20y%3D%2216%22%3E999%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22128.5%22%20y%3D%2216%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22140.5%22%20y%3D%2216%22%3E...%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22152.5%22%20y%3D%2216%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22173.5%22%20y%3D%2216%22%3E999%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22190.5%22%20y%3D%2216%22%3E..%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22197.5%22%20y%3D%2216%22%3E9%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22165.5%22%20y%3D%2223%22%3E%26%23xF300%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22169.5%22%20y%3D%2223%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22173.5%22%20y%3D%2223%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22177.5%22%20y%3D%2223%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22181.5%22%20y%3D%2223%22%3E%26%23xF304%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22184.5%22%20y%3D%2223%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22188.5%22%20y%3D%2223%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22192.5%22%20y%3D%2223%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22196.5%22%20y%3D%2223%22%3E%26%23xF301%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22164.5%22%20y%3D%2232%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22177.5%22%20y%3D%2232%22%3Es%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22183.5%22%20y%3D%2232%22%3E%26%23x1ED1%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22196.5%22%20y%3D%2232%22%3E9%3C%2Ftext%3E%3C%2Fsvg%3E)
b) format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A'math1b5dc6aeb84bcea29a76d1e9d00'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAAERjdnQgDVUNBwAAAWAAAAA6Z2x5ZoPi2VsAAAGcAAABcWhlYWQQC2qxAAADEAAAADZoaGVhCGsXSAAAA0gAAAAkaG10eE2rRkcAAANsAAAAEGxvY2EAHTwYAAADfAAAABRtYXhwBT0FPgAAA5AAAAAgbmFtZaBxlY4AAAOwAAABn3Bvc3QB9wD6AAAFUAAAACBwcmVwa1uragAABXAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADAAAAAIAAgAAgAAACsALgA9%2F%2F8AAAArAC4APf%2F%2F%2F9b%2F1P%2FGAAEAAAAAAAAAAAAAAVQDLACAAQAAVgAqAlgCHgEOASwCLABaAYACgACgANQAgAAAAAAAAAArAFUAgACrANUBAAErAAcAAAACAFUAAAMAA6sAAwAHAAAzESERJSERIVUCq%2F2rAgD%2BAAOr%2FFVVAwAAAQCAAFUC1QKrAAsASQEYsgwBARQTELEAA%2FaxAQT1sAo8sQMF9bAIPLEFBPWwBjyxDQPmALEAABMQsQEG5LEBARMQsAU8sQME5bELBfWwBzyxCQTlMTATIREzESEVIREjESGAAQBVAQD%2FAFX%2FAAGrAQD%2FAFb%2FAAEAAAEAIAAAAKAAgAADAC8YAbAEELAD1LADELAC1LADELAAPLACELABPACwBBCwA9SwAxCwAjywABCwATwwMTczFSMggICAgAACAIAA6wLVAhUAAwAHAGUYAbAIELAG1LAGELAF1LAIELAB1LABELAA1LAGELAHPLAFELAEPLABELACPLAAELADPACwCBCwBtSwBhCwB9SwBxCwAdSwARCwAtSwBhCwBTywBxCwBDywARCwADywAhCwAzwxMBMhNSEdASE1gAJV%2FasCVQHAVdVVVQAAAAABAAAAAQAA1XjOQV8PPPUAAwQA%2F%2F%2F%2F%2F9Y6E3P%2F%2F%2F%2F%2F1joTcwAA%2FyAEgAOrAAAACgACAAEAAAAAAAEAAAPo%2F2oAABdwAAD%2FtgSAAAEAAAAAAAAAAAAAAAAAAAAEA1IAVQNWAIAAyAAgA1YAgAAAAAAAAAAoAAAAoQAAAOcAAAFxAAEAAAAEAF4ABQAAAAAAAgCABAAAAAAABAAA3gAAAAAAAAAVAQIAAAAAAAAAAQASAAAAAAAAAAAAAgAOABIAAAAAAAAAAwAwACAAAAAAAAAABAASAFAAAAAAAAAABQAWAGIAAAAAAAAABgAJAHgAAAAAAAAACAAcAIEAAQAAAAAAAQASAAAAAQAAAAAAAgAOABIAAQAAAAAAAwAwACAAAQAAAAAABAASAFAAAQAAAAAABQAWAGIAAQAAAAAABgAJAHgAAQAAAAAACAAcAIEAAwABBAkAAQASAAAAAwABBAkAAgAOABIAAwABBAkAAwAwACAAAwABBAkABAASAFAAAwABBAkABQAWAGIAAwABBAkABgAJAHgAAwABBAkACAAcAIEATQBhAHQAaAAgAEYAbwBuAHQAUgBlAGcAdQBsAGEAcgBNAGEAdABoAHMAIABGAG8AcgAgAE0AbwByAGUAIABNAGEAdABoACAARgBvAG4AdABNAGEAdABoACAARgBvAG4AdABWAGUAcgBzAGkAbwBuACAAMQAuADBNYXRoX0ZvbnQATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlAAADAAAAAAAAAfQA%2BgAAAAAAAAAAAAAAAAAAAAAAAAAAuQcRAACNhRgAsgAAABUUE7EAAT8%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3ES%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2212%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2215.5%22%20y%3D%2221%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2228.5%22%20y%3D%2216%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2241.5%22%20y%3D%2216%22%3E8%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2253.5%22%20y%3D%2216%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2270.5%22%20y%3D%2216%22%3E88%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2286.5%22%20y%3D%2216%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22107.5%22%20y%3D%2216%22%3E888%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22128.5%22%20y%3D%2216%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22140.5%22%20y%3D%2216%22%3E...%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22152.5%22%20y%3D%2216%22%3E%2B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22170.5%22%20y%3D%2216%22%3E88%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1b5dc6aeb84bcea29a76d1e9d00%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22183.5%22%20y%3D%2216%22%3E...%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22192.5%22%20y%3D%2216%22%3E8%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22163.5%22%20y%3D%2223%22%3E%26%23xF300%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22167.5%22%20y%3D%2223%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d356305d8b17%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%22171.5%22%20y%3D%2223%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal24eedbba2d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Lời giải

3. Bài tập vận dụng
Bài 1: Cho cấp số nhân (un) có các số hạng khác không, tìm u1 biết: 
Lời giải:

Bài 2: Tìm tổng 5 số hạng đầu tiên của cấp số nhân, biết 
Lời giải:

Bài 3: Một cấp số nhân dương có 4 số hạng, công bội q bằng 1/4 lần số hạng thứ nhất, tổng của hai số hạng đầu bằng 24. Tìm tích các số hạng cấp số nhân đó?
Lời giải:
Gọi 4 số lập thành cấp số cộng là u1,u2,u3,u4
u1=8,u2=16,u3=32,u4=64. Khi đó tích cần tìm là: 8.6.32.64 = 98304.
Bài 4: Cho bốn số nguyên biết rằng ba số hạng đầu lập thành một cấp số nhân, ba số hạng sau lập thành một cấp số cộng. Tổng của hai số hạng đầu và cuối bằng 14, còn tổng hai số ở giữa bằng 12. Tổng của bốn số nguyên đó là?
Lời giải:
Gọi 4 số cần tìm là a,b,c,d. Dựa vào giả thiết ta có hệ:

Vậy tổng 4 số nguyên đó là: 2 + 4 + 8 +12 = 26.
Bài 5: Cho cấp số nhân có 7 số hạng, số hạng thứ tư bằng 6 và số hạng thứ 7 gấp 243 lần số hạng thứ hai. Hãy tìm số hạng còn lại của CSN đó.
Lời giải:
Từ giả thiết ta có

Vậy u1=2/9,u2=2/3,u3=2,u4=6,u5=18,u6=54,u7=162.
4. Bài tập tự luyện
Bài 1: Trong các dãy số sau, dãy số nào là cấp số nhân?

Lời giải:
Đáp án: C
Đáp án là C (CSN với công bội là 7).
Bài 2: Cho cấp số nhân (un) có u1=5; u2 = 8. Tìm u4
A. 512/25
B. 125/512
C. 625/512
D. 512/125
Lời giải:
Đáp án: A

Đáp án A
Bài 3: Cho một cấp số nhân có 5 số hạng với công bội dương. Biết rằng số hạng thứ hai bằng 3, số hạng thứ tư bằng 6. Tính tổng của cấp số nhân đó?

Lời giải:
Đáp án: C
Kí hiệu u1,u2,u3,u4,u5 là các số hạng của cấp số nhân

Đáp án C.
Bài 4: Cho tam giác ABC cân (AB=AC), có cạnh đáy BC, đường cao AH, cạnh bên AB theo thứ tự số đo lập thành một cấp số nhân. Hãy tính công bội q của cấp số nhân đó.

Lời giải:
Đáp án: C
Theo giả thiết AB=AC, BC,AH,AB lập thành cấp số nhân nên ta có hệ:

Từ đó ta có kết quả sau: 2cotC = sinC ⇔ 2cosC =sin2C = 1-cos2C ⇔ cos2C + 2cosC -1 =0 ⇒cosC = -1 +√2 (0º< C < 90º)
Do C là góc nhọn nên 
Cho nên công bội của cấp số nhân là:

Đáp án C.
Bài 5: Tìm các số (x,y) biết y < 0 và các số x+6y, 5x+2y, 8x+y theo thứ tự lập thành cấp số cộng đồng thời các số x+ 5/3, y -1, 2x – 3y theo thứ tự đó lập thành một cấp số nhân.
A. (3, -1)
B. (-3, -1)
C. (-1,-3)
D. (-1,3)
Lời giải:
Đáp án: B
Ta có hệ phương trình:

Từ đó ta suy ra

Thế (1) vào (2) ta được: 8y2+7y-1=0⇒y=-1 hoặc y=1/8
Do y < 0 , ta được y = -1, x = -3
Đáp án B.
Bài 6: Trong các dãy số sau dãy số nào là cấp số nhân?
A. 1,3,5,7,9
B. -1,-3,1,3,5
C. 1,2,4,16,256
D. 1,2,4,8,16
Lời giải:
Đáp án: D
Đáp án D (CSN với công bội là 2).
Bài 7: Trong các dãy số sau, dãy nào là dãy số nhân?
Lời giải:
Đáp án: C
Đáp án C vì 
Bài 8: Số hạng đầu tiên của cấp số nhân (un) thoả mãn hệ 
A. 2 B. 12 C. 24 D. 0
Lời giải:
Đáp án: B
Đáp án B.
Bài 9: Cho ba số tạo thành một cấp số nhân mà tổng của chúng bằng 93. Ta có thể sắp xếp chúng (theo thứ tự của cấp số nhân kế trên) như là số hạng thứ nhất, thứ hai và thứ bảy của một cấp số cộng. Tìm tích của 3 số đó.
A. 3375
B. 64
C. 2744
D. 1000
Lời giải:
Đáp án: A
Gọi 3 số cần tìm là a, b, c. Vì 3 số trên là số hạng thứ nhất, thứ hai và thứ bảy của một cấp số cộng nên ta có
(với d là công sai của cấp số cộng đó).
3 số trên tạo thành cấp số nhân với tổng là 93 nên ta có:
Vậy a = 3, b = 15, c = 75. Tích ba số trên là 3.15.75 = 3375.
Đáp án A.
Bài 10: Tìm số hạng đầu của cấp số nhân có bốn số hạng, biết tổng ba số hạng đầu bằng
, đồng thời theo thứ tự, chúng là số hạng thứ nhất, thứ tư và thứ tám của một cấp số cộng.
A. 4
B. 16/9
C. 2/3
D. -1
Lời giải:
Đáp án: A
Gọi 4 số cần tìm là a, b, c, d. Vì 3 số hạng đầu là số hạng thứ nhất, thứ tư và thứ tám của một cấp số cộng nên ta có
(với d là công sai của cấp số cộng đó).
Từ giả thiết ta có:
Vậy số hạng đầu là 4. Đáp án A.
Bài 11: Cho 3 số x, 3, y lập thành một cấp số nhân và x4=y√3. Tìm công bội q của cấp số đó
A. 1/3
B. √3
C. 3
D. 1/√3
Lời giải:
Đáp án: B
Vậy công bội q = √3.
Đáp án B.
Bài 12: Cho tam giác ABC có các cạnh tương ứng a,b,c. Biết
theo thứ tự lập thành một cấp số nhân. Tìm số đo góc B.
A. 30º
B. 45º
C. 15º
D. 60º
Lời giải:
Đáp án: D
Vì
theo thứ tự lập thành một cấp số nhân và tam giác ABC vuông nên ta có:
Đáp án D.
Bài 13: Giả sử a,b,c ,d lập thành một cấp số nhân. Tính giá trị biểu thức
(a-c)2+(b-c)2+(b-d)2-(a-d)2
A. 3
B. 2
C. 1
D. 0
Lời giải:
Đáp án: D
(a-c)2+(b-c)2+(b-d)2-(a-d)2=2b2+2c2-2ac-2bc-2bd+2ad
Vì a, b, c, d là một cấp số nhân nên ta có: b2=ac,b2=bd.
Khi đó ta có: (a-c)2+(b-c)2+(b-d)2-(a-d)2=2ad-2bc=0.
Đáp án D.
Bài 14: Công bội nguyên dương của cấp số nhân (un) thoả mãn 
A. 3 B. 2 C. 1 D. 0
Lời giải:
Đáp án: B
(do dãy trên có công bội nguyên dương)
Đáp án B.
Bài 15: Cho dãy số (un) với
. Khẳng định nào sau đây đúng?
A. (un) không phải là cấp số nhân.
B. (un) là cấp số nhân có công bội q = 5 và số hạng đầu u1=3/2
C. (un) là cấp số nhân có công bội q = 5 và số hạng đầu u1=15/2
D. (un) là cấp số nhân có công bội q = 2.5 và số hạng đầu u1=3
Lời giải:
Đáp án: C
Vậy công bội q = 5, số hạng đầu là: 15/2.
Đáp án C.
Xem các Phương pháp giải hay, chi tiết khác:
Công thức cấp số cộng
Công thức tính công sai của cấp số cộng
Công thức tìm số hạng tổng quát của cấp số cộng
Công thức tính tổng n số hạng đầu của cấp số cộng
Công thức cấp số nhân