Question

数列{an}满足a1=1/2,a1+a2+a3+…+an=n^2×an

数列{an}满足a1=1/2,a1+a2+a3+…+an=n^2×an (1)求a2，a3，a4的值; (2)求数列{an}的通项 要详细过程!

Answer 1

1、a1+a2=4a2，所以a2=a1/3=1/6
a1+a2+a3=9a3，所以a3=(a1+a2)/8=1/12
a1+a2+a3+a4=16a4，所以a4=(a1+a2+a3)/15=1/20
2、油a1、a2、a3、a4
猜想其an=1/n-1/(n+1)
则a1+a2+a3+…+an
=1-1/2+1/2-1/3+……+1/(n-1)-1/n+1/n-1/(1+n)
=1-1/(n+1)=n/(n+1)
n^2×an
=n²(1/n-1/(n+1))
=n/(n+1)
即a1+a2+a3+…+an=n^2×an
满足题意，所以an=1/n-1/(n+1)