已知数列｛xn｝的通项xn=2^n+np(n为正整数,p为常数）且x1,x4,x5成等差数列,求

题目

已知数列｛xn｝的通项xn=2^n+np(n为正整数,p为常数）且x1,x4,x5成等差数列,求
求（1）p的值 （2）数列｛xn｝的前n项和

答案

x1=2^1+p=2+p
x4=2^4+4p=16+4p
x5=2^5+5p=32+5p
2x4=x1+x5
2(16+4p)=2+p+32+5p
32+8p=6p+34
2p=2
p=1
xn=2^n+np
xn=2^n+n
Sn=x1+x2+x3+x4+.+xn
=2^1+1+2^2+2+2^3+3+.+2^n+n
=2^1+2^2+2^3+.+2^n+1+2+3+.+n
=2(1-2^n)/(1-2)+(1+n)n/2
=2(2^n-1)+(1+n)n/2
=2^(n+1)-2+(1+n)n/2