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Câu 49 trang 124 SGK Đại số và Giải tích 11 Nâng cao

Cho dãy hình vuông H1, H2, …, Hn,…
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Cho dãy hình vuông H1, H2, …, Hn,… Với mỗi số nguyên dương n, gọi un, pn và Sn lần lượt là độ dài cạnh, chu vi và diện tích của hình vuông Hn.

LG a

Giả sử dãy số (un) là một cấp số cộng với công sai khác 0. Hỏi khi đó các dãy số (pn) và (Sn) có phải là các cấp số cộng hay không ? Vì sao ?

Lời giải chi tiết:

Theo giả thiết ta có : \({p_n} = 4{u_n}\text{ và }{S_n} = u_n^2\) với mọi \(n \in N^*\)

Gọi d là công sai của cấp số cộng (un) , d ≠ 0. Khi đó với mọi \(n \in N^*\), ta có :

\({p_{n + 1}} - {p_n}  = 4{u_{n + 1}} - 4{u_n}\) \(= 4\left( {{u_{n + 1}} - {u_n}} \right) = 4d\) (không đổi)

Vậy (pn) là cấp số cộng.

\({S_{n + 1}} - {S_n}  = u_{n + 1}^2 - u_n^2\) \(= \left( {{u_{n + 1}} - {u_n}} \right)\left( {{u_{n + 1}} + {u_n}} \right) \) \(= d\left( {{u_{n + 1}} + {u_n}} \right)\) không là hằng số (do d ≠ 0)

Vậy (Sn) không là cấp số cộng.

LG b

Giả sử dãy số (un) là một cấp số nhân với công bội dương. Hỏi khi đó các dãy số (pn) và (Sn) có phải là các cấp số nhân hay không ? Vì sao ?

Lời giải chi tiết:

Gọi q là công bội của cấp số nhân (unꦫ), q > 0. Khi đó với mọi \(n \in N^*\), ta có :

\({{{p_{n + 1}}} \over {{p_n}}} = {{4{u_{n + 1}}} \over {4{u_n}}} = q\) (không đổi) \({{{S_{n + 1}}} \over {{S_n}}} = {{u_{n + 1}^2} \over {u_n^2}}  = {\left( {\frac{{{u_{n + 1}}}}{{{u_n}}}} \right)^2}= {q^2}\) (không đổi)

Từ đó suy ra các dãy số (pn) và (Sn) là cấp số nhân.

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