问高一数列题目

来源：百度知道 编辑：UC知道 时间：2024/09/18 08:40:26
1.在数列{An}中，若A1=1,A(n+1)=2An+3(n>=1),求该数列的通项An
2.等比数列{An}前n项的和为2^n -1,求数列{A^2 n}前n项的和

A(1) = 1,

A(n+1) = 2A(n) + 3,

A(n+1) + 3 = 2[A(n) + 3]

{A(n)+3} 是首项为A(1)+3 = 4,公比为2的等比数列。
A(n) + 3 = 4*2^(n-1)
A(n) = 4*2^(n-1) - 3. n = 1,2,...

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2,
A(n) = aq^(n-1),n = 1,2,...

A(1) = 2-1 = 1,
A(2) = S(2) - S(1) = 2^2 - 1 - 1 = 2,
q = A(2)/A(1) = 2/1 = 2,

A(n) = 2^(n-1), n = 1,2,....

[A(n)]^2 = 4^(n-1)

[A(1)]^2 + [A(2)]^2 + ... + [A(n)]^2 = 1 + 4 + ... + 4^(n-1) = [4^n - 1]/[4-1] = [4^n - 1]/3.