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Câu 57 trang 177 SGK Đại số và Giải tích 11 Nâng cao

Cho một cấp số nhân (un), trong đó
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Cho một cấp số nhân (un), trong đó

\(243{u_8} = 32{u_3}\,\text{ với }\,{u_3} \ne 0.\)

LG a

Tính công bội của cấp số nhân đã cho.

Phương pháp giải:

Sử dụng công thức \({u_{n}} = {u_1}{q^{n-1}}\)

Lời giải chi tiết:

Cấp số nhân đã cho có số hạng đầu là u1,công bội q.

Ta có: u3 = u1.q2 ≠ 0 ⇒ u1 ≠ 0; q ≠ 0

Theo giả thiết ta có: 243 u8 = 32.u3 nên:

243.u1.q7 = 32.u1.q2

243.u1.q7 - 32.u1.q2 = 0

u1.q2. (243.q5 - 32) = 0

243.q5 - 32 = 0 ( vì u1 ≠ 0; q ≠ 0 )

\( \Leftrightarrow {q^5} = \frac{{32}}{{243}} = {\left( {\frac{2}{5}} \right)^5}\)\( \Leftrightarrow q = \frac{2}{5}\)

Cách khác:

Ta có: \({u_8} = {u_3}{q^5}\) với q là công bội của cấp số nhân. Thay vào đẳng thức đã cho, ta được : \(243{u_3}{q^5} = 32{u_3}\)

Vì u3🦩≠ 0 nên :  \({q^5} = {{32} \over {243}} = {\left( {{2 \over 3}} \right)^5}\) \( \Leftrightarrow q = {2 \over 3}\)

LG b

Biết rằng tổng của cấp số nhân đã cho bằng \({3^5},\) tính u1.

Phương pháp giải:

Công thức tổng cấp số nhân lùi vô hạn \(S = \dfrac{{{u_1}}}{{1 - q}}\)

Lời giải chi tiết:

Tổng của cấp số nhân lùi vô hạn đó là \(S = {{{u_1}} \over {1 - q}}.\) Từ đó, ta có : \({3^5} = {{{u_1}} \over {1 - {2 \over 3}}}\) \(\Leftrightarrow {3^5} = \frac{{{u_1}}}{{1/3}} \) \(\Leftrightarrow {u_1} = {3^5}.\frac{1}{3} = {3^4} = 81\)

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