已知an=2n-1，bn=2^n,记Cn=anbn,求数列Cn的前n项和?????？？？？？？

Question

如题所述

Answer 1

记S=∑Cn=1×2+3×2^2+5×2^3+.............+(2n-1)×2^n
那么2S=1×2^2+3×2^2+5×2^4+.............+(2n-1)×2^(n+1)
两式相减得 S=1×2-2×2^2-2×2^3-......-2×2^n +(2n-1)2^(n+1)
= 2 -2^3[1+2+....+2^(n-2)]+(2n-1)2^(n+1)
=2-2^3[2^(n-1)-1]+(2n-1)2^(n+1)
=(2n-1)2^(n+1)-2^(n+2)-6
=(2n-3)2^(n+1)-6

Answer 2

{(2n-1)/2^n}= 2n/2^n - 1/2^n

对于后一部分 1/2^n , 其前n项和为等比数列求和
S2 = 1/2 + 1/2^2 + 1/2^3 + …… 1/2^n
= (1/2) * [1 - (1/2)^n]/(1 - 1/2)
= 1 - 1/2^n

对于前一部分 2n/2^n
S1 = 2*(1/2 + 2/2^2 + 3/2^3 + …… + n/2^n)
两端乘2
2S1 = 2 * [1 + 2/2 + 3/2^2 + …… + n/2^(n-1)]
两式相减, 将分母方次相同的项凑在一起
2S1 - S1 = S1
= 2*{ 1 + （2/2 - 1/2）+ (3/2^2 - 2/2^2) + …… + [n/2^(n-1) - (n-1)/2^(n-1 ) - n/2^n }
= 2 * [1 + 1/2 + 1/2^2 + 1/2^(n-1) - n/2^n]
= 2 * { 1 * [1 - (1/2)^n]/(1 -1/2) - n/2^n}
= 2 * [2 - 1/2^(n-1) - n/2^n]
= 4 - 4/2^n - 2n/2^n

S = S1 - S2
= 4 - 4/2^n - 2n/2^n - 1 + 1/2^n
= 3 - (3 + 2n)/2^n