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Câu 29 trang 120 SGK Đại số và Giải tích 11 Nâng cao

Trong các dãy số dưới đây
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Trong các dãy số dưới đây, dãy số nào là cấp số nhân ? Hãy xác định công bội của cấp số nhân đó.

LG a

Dãy số \(1, -2, 4, -8, 16, -32, 64\)

Lời giải chi tiết:

Dãy số đã cho là một cấp số nhân với công bội \(q = -2\).

LG b

Dãy số (un) với \({u_n} = n{.6^{n + 1}}\)

Lời giải chi tiết:

\(\frac{{{u_{n + 1}}}}{{{u_n}}} = \frac{{\left( {n + 1} \right){6^{n + 1}}}}{{n{{.6}^n}}} = \frac{{6\left( {n + 1} \right)}}{n}\) với mọi \(n ≥ 1\).

Do \(\frac{{6\left( {n + 1} \right)}}{n}\) không phải là hằng số nên (un) không phải là cấp số nhân.

LG c

Dãy số (vn൲) với \({v_n} = {\left( { - 1} \right)^n}{.3^{2n}}\)

Lời giải chi tiết:

\({{{v_{n + 1}}} \over {{v_n}}} = {{{{\left( { - 1} \right)}^{n + 1}}{{.3}^{2\left( {n + 1} \right)}}} \over {{{\left( { - 1} \right)}^n}{{.3}^{2n}}}} = \frac{{ - {{1.3}^{2n + 2}}}}{{{3^{2n}}}} = - 9\) với mọi \(n ≥ 1\).

Suy ra (vn) là một cấp số nhân với công bội \(q = -9\).

LG d

Dãy số (xn꧃) với \({x_n} = {\left( { - 4} \right)^{2n + 1}}\) .

Lời giải chi tiết:

\({{{x_{n + 1}}} \over {{x_n}}} = {{{{\left( { - 4} \right)}^{2n + 3}}} \over {{{\left( { - 4} \right)}^{2n + 1}}}}  = \frac{{{{\left( { - 4} \right)}^{2n + 1}}.{{\left( { - 4} \right)}^2}}}{{{{\left( { - 4} \right)}^{2n + 1}}}}= 16\) với mọi \(n ≥ 1\).

Suy ra (xn) là một cấp số nhân với công bội \(q = 16\).

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