Question

数列an=n乘2的n次方,求sn

Answer 1

S[n] = 1·2 + 2·2^2 + 3·2^3 + ...+ n·2^n

2·S[n] = 1·2^2 + 2·2^3 + 3·2^4 + ...+ n·2^(n+1)

两式相减

-S[n] = S[n] - 2·S[n] = 1·2 + 2^2 + 2^3 + ...+ 2^n - n·2^(n+1)

= 2^(n+1) - 2 - n·2^(n+1)

= (1-n)·2^(n+1) - 2

所以

S[n] = (n-1)·2^(n+1) + 2