数学红对勾9:已知数列{an}满足a1=1/2,an=an-1+1/(n^2-1) (n≧2),则an的通项公式.

问题描述:

数学红对勾9:已知数列{an}满足a1=1/2,an=an-1+1/(n^2-1) (n≧2),则an的通项公式.

an=a(n-1)+1/2(1/(n-1) -1/(n+1))
a(n-1)=a(n-2)+1/2(1/(n-2)-1/n)
.
a2=a1+1/2(1/1-1/2)
上面(n-1)式相加得
an=a1+1/2(1/(n-1)-1/(n+1) +1/(n-2)-1/n +.+1/2-1/3+1/1-1/2)
=1/2+1/2(1-1/(n+1))
=1-1/2*1/(n+1)