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Câu 16 trang 109 SGK Đại số và Giải tích 11 Nâng cao

Cho dãy số (un) xác định bởi
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Cho dãy số (un) xác định bởi

\({u_1} = 1\,\text{ và }\,{u_{n + 1}} = {u_n} + \left( {n + 1} \right){.2^n}\) với mọi \(n ≥ 1\)

LG a

Chứng minh rằng (un) là một dãy số tăng.

Lời giải chi tiết:

Từ hệ thức xác định dãy số (un), ta có:

\({u_{n + 1}} - {u_n} = \left( {n + 1} \right){.2^n} > 0\;\forall n \ge 1.\)

Do đó (un) là một dãy số tăng.

LG b

Chứng minh rằng \({u_n} = 1 + \left( {n - 1} \right){.2^n}\) với mọi \(n ≥ 1\).

Lời giải chi tiết:

Ta sẽ chứng minh \({u_n} = 1 + \left( {n - 1} \right){.2^n}\)  (1) với mọi \(n ≥ 1\), bằng phương pháp qui nạp. +) Với \(n = 1\), ta có \({u_1} = 1 = 1 + \left( {1 - 1} \right){.2^1}.\) Như vậy (1) đúng khi \(n = 1\) +) Giả sử (1) đúng khi \(n = k, k \in\mathbb N^*\), tức là: \({u_k} = 1 + \left( {k - 1} \right){2^k}\) +) Ta sẽ chứng minh (1) cũng đúng với \(n = k + 1\).

Thật vậy, từ hệ thức xác định dãy số (un) và giả thiết qui nạp, ta có :

\({u_{k + 1}} = {u_k} + \left( {k + 1} \right){.2^k} \) \(= 1 + \left( {k - 1} \right){.2^k} + \left( {k + 1} \right){.2^k} \) \( = 1 + k{.2^k} - {2^k} + k{.2^k} + {2^k} \) \(= 1 + 2k{.2^k}= 1 + k{.2^{k + 1}}\) Vậy (1) đúng với mọi \(n ≥ 1\).

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