已知数列{an}的通项为an={n(n为奇数)2^n/2(n为偶数),Sn为数列{an}的前n项和,则S20

Question

已知数列{an}的通项为an={n(n为奇数) 2^n/2(n为偶数),Sn为数列{an}的前n项和,则S20

Answer 1

a(2n-1)=2n-1,
t(n)=a(1)+a(3)+...+a(2n-1)=n(n+1)-n=n^2.
a(2n)=2^n,
u(n)=a(2)+a(4)+...+a(2n)=2[1+2+...+2^(n-1)]=2[2^n - 1]/(2-1) = 2[2^n-1]=2^(n+1) - 2.
s(20)=a(1)+a(3)+...+a(19)+a(2)+a(4)+...+a(20)
=t(10)+u(10)=10^2 + 2^(11)- 2
=100 + 2048 - 2
= 2146