1/(1x2x3)+1/(2x3x4)+1/(3x4x5)+.1/(nx(n+1)x(n+2)=?

题目

1/(1x2x3)+1/(2x3x4)+1/(3x4x5)+.1/(nx(n+1)x(n+2)=?
请问这个题目该怎么思考,我经常碰到,可一碰到新的就无从下手.请高手教教我方法

答案

=1/2×[1/1×2-1/2×3+1/2×3-1/3×4+……+1/n(n+1)-1/(n+1)(n+2)]
=1/2×[1/1×2-1/(n+1)(n+2)]
=1/4-2/(4n²+12n+8)
=(n²+3n)/(4n²+12n+8)