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Câu 41 trang 122 SGK Đại số và Giải tích 11 Nâng cao

Số hạng thứ hai
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Đề bài

Số hạng thứ hai, số hạng đầu và số hạng thứ ba của một cấp số cộng với công sai khác 0 theo thứ tự đó lập thành một cấp số nhân. Hãy tìm công bội của cấp số nhân đó.

Phương pháp giải - Xem chi tiết

Tính chất CSC: \[{u_{k + 1}} + {u_{k - 1}} = 2{u_k}\] Số hạng TQ của CSN: \[{u_n} = {u_1}{q^{n - 1}}\]

Lời giải chi tiết

Kí hiệu (un) là cấp số cộng đã cho và gọi q là công bội của cấp số nhân u2, u1, u3.

Vì cấp số cộng (un) có công sai khác 0 nên các số u1, u2, u3ꦕ đôi một khác nhau, suy ra \(q \ne 0,q \ne 1,{u_2} \ne 0\)

Do u2, u1, u3 là CSN nên u1 = u2q, u3 = u2q2

Do u1, u2, u3 là CSC nên:

u1 + u3 = 2u2

\( \Rightarrow {u_2}q + {u_2}{q^2} = 2{u_2}\)\( \Leftrightarrow {u_2}\left( {q + {q^2}} \right) = 2{u_2} \)\(\Leftrightarrow {q^2} + q - 2 = 0\,\left( {\text{vì }\,{u_2} \ne 0} \right) \)

\( \Leftrightarrow \left[ \begin{array}{l}
q = 1\left( {loai} \right)\\
q = - 2\left( {TM} \right)
\end{array} \right.\)

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