一个数列{an},当N为奇数时,an=5n+1,当n为偶数时,an=2^(n...

题目

一个数列{an},当N为奇数时,an=5n+1,当n为偶数时,an=2^(n...
一个数列{an},当N为奇数时,an=5n+1,当n为偶数时,an=2^(n/2),求这 个数列的前N项和

答案

当n=2k时
Sn=前k奇数项的和+前k偶数项的和
=(1+5k+1)*k/2+2(1-2^k)/(1-2)=(5k+2)k/2+2(2^k-1)
当n=2k+1时
Sn=前k+1奇数项的和+前k偶数项的和
=(1+5(k+1)+1)*(k+1)/2+2(1-2^k)/(1-2)=(5k+7)(k+1)/2+2(2^k-1)