已知{an}的通项an=(2n-3)*4^n-2 求数列{an}的前n项和Sn
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a1 = -1 * 4^(-1)
a2 = 1 * 4^0
a3 = 3 * 4^1
a4 = 5 * 4^2
......
an = (2n - 3) * 4^(n - 2)
Sn = a1 + a2 + a3 + a4 + a5 +......+ (2n - 3) * 4^(n - 2)
Sn = -1 * 4^(-1) + 1 * 4^0 + 3 * 4^1 + 5 * 4^2+......+ (2n - 3) * 4^(n - 2)----(1)
等式两边同时乘以 4 得
4Sn = -1 * 4^0 + 1 * 4^1 + 3 * 4^2 + 5 * 4^3+......+ (2n - 3) * 4^(n - 1)——--(2)
(1)-(2)得
-3Sn = -1 * 4^(-1) + 2 * 4^0 + 2 * 4^1 + 2 * 4^2 +......+ 2 * 4^(n - 2) - (2n - 3) * 4^(n - 1)
= 2 [ 4^0 + 4^1 + 4^2 +......+ 4^(n - 2)] - (2n - 3) * 4^(n - 1) -1/4
= 2 [4^(n -1) - 1] /3 - (2n - 3) * 4^(n - 1) -1/4
= 4^(n -1) * [ 2 - 3 (2n - 3) ] /3 - 2/3 - 1/4
= 4^(n -1) * (11 - 6n) /3 - 17/12
Sn = 4^(n -1) * (6n - 11) /9 + 17/36
a2 = 1 * 4^0
a3 = 3 * 4^1
a4 = 5 * 4^2
......
an = (2n - 3) * 4^(n - 2)
Sn = a1 + a2 + a3 + a4 + a5 +......+ (2n - 3) * 4^(n - 2)
Sn = -1 * 4^(-1) + 1 * 4^0 + 3 * 4^1 + 5 * 4^2+......+ (2n - 3) * 4^(n - 2)----(1)
等式两边同时乘以 4 得
4Sn = -1 * 4^0 + 1 * 4^1 + 3 * 4^2 + 5 * 4^3+......+ (2n - 3) * 4^(n - 1)——--(2)
(1)-(2)得
-3Sn = -1 * 4^(-1) + 2 * 4^0 + 2 * 4^1 + 2 * 4^2 +......+ 2 * 4^(n - 2) - (2n - 3) * 4^(n - 1)
= 2 [ 4^0 + 4^1 + 4^2 +......+ 4^(n - 2)] - (2n - 3) * 4^(n - 1) -1/4
= 2 [4^(n -1) - 1] /3 - (2n - 3) * 4^(n - 1) -1/4
= 4^(n -1) * [ 2 - 3 (2n - 3) ] /3 - 2/3 - 1/4
= 4^(n -1) * (11 - 6n) /3 - 17/12
Sn = 4^(n -1) * (6n - 11) /9 + 17/36
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