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Câu 43 trang 122 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 và un + 1 = 5un + 8 với mọi n ≥ 1.

LG a

Chứng minh rằng dãy số (vn), với vn = un + 2, là một cấp s🌸ố nhân. Hãy tìm số hạng tổng quát c꧃ủa cấp số nhân đó.

Phương pháp giải:

Cộng cả hai vế của đẳng thức đã cho với 2 để làm xuất hiện \(v_{n+1}\) và \(v_n\)

Lời giải chi tiết:

Với mọi n ≥ 1, ta có : \({u_{n + 1}} = 5{u_n} + 8\) \(\Rightarrow {u_{n + 1}} + 2 = 5{u_n} + 10 \) \(\Leftrightarrow {u_{n + 1}} + 2 = 5\left( {{u_n} + 2} \right) \) \(\Rightarrow {v_{n + 1}} = 5{v_n}\)

Do đó (vn) là một cấp số nhân với số hạng đầu \({🦄v_1} = {\rm{ }}{u_1} + {\rm{ }}2{\rm{ }} = {\rm{🌳 }}3\) và công bội q = 5.

Số hạng tổng quát : \({v_n} = {\rm{ }}{3.5^{n{\rm{ }}-{\rm{ }}1}}\)

LG b

Dựa vào kết quả phần a, hãy tìm số hạng tổng quát của dãy số (un).

Phương pháp giải:

Sử dụng mối quan hệ giữa \(v_n\) và \(u_n\) kết hợp với số hạng TQ đã tìm được ở câu a để suy ra \(u_n\).

Lời giải chi tiết:

\({v_n} = {u_n} + 2 \) \(\Rightarrow {u_n} = {v_n} - 2 = {3.5^{n - 1}} - 2\) với mọi \(n ≥ 1\)

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