已知数列an中a1=1/2且a(n+1)-an=1/(4n^2-1),求数列an

题目

已知数列an中a1=1/2且a(n+1)-an=1/(4n^2-1),求数列an

答案

a(n+1)
=an+1/(4n^2-1)
=an+1/(2n+1)(2n-1)
=an+1/2*[1/(2n-1)-1/(2n+1)],
则
an=a(n-1)+1/2*[1/(2n-3)-1/(2n-1)],
a(n-1)=a(n-2)+1/2*[1/(2n-5)-1/(2n-3)],
...
a3=a2+1/2*(1/3-1/5)
a2=a1+1/2*(1/1-1/3),
两边相加,消去中间项得
an=a1+1/2*[1-1/(2n-1)]
=1-1/[2(2n-1)].