Question

已知数列an=1/n(n+2），求Sn.

要有详细过程，谢谢

Answer 1

Sn=1/3+1/(2x4)+1/(3x5)+......+1/[n(n+2)]
={2/3+2/(2x4)+2/(3x5)+......+2/[n(n+2)]}/2
={(1-1/3)+(1/2-1/4)+(1/3-1/5)+......+[1/n-1/(n+2)]}/2
={1-1/3+1/2-1/4+1/3-1/5+......+1/n-1/(n+2)}/2
={1+1/2-1/(n+1)-1/(n+2)}/2
={3/2-(2n+3)/[(n+1)(n+2)]}/2
={[3(n+1)(n+2)-2(2n+3)]/[2(n+1)(n+2)]}/2
={[3(n²+3n+2)-(4n+6)]/[2(n+1)(n+2)]}/2
={[3n²+9n+6-4n-6]/[2(n+1)(n+2)]}/2
={[3n²+5n]/[2(n+1)(n+2)]}/2
=n(3n+5)/[4(n+1)(n+2)]

追问

能纸上写出来吗

Answer 2

an
= 1/[n(n+2)]
=(1/2)[ 1/n - 1/(n+2) ]
Sn
=a1+a2+...+an
=(1/2) [ 1 + 1/2 - 1/(n+1) - 1/(n+2) ]
=(1/2) [ 3/2 - 1/(n+1) - 1/(n+2) ]

Answer 3

用裂项相消法

追答

拆开后消掉就是