求数列1×2分之6,2×3分之6,3×4分之6…n×n+1分之6,的前n项和

题目

求数列1×2分之6,2×3分之6,3×4分之6…n×n+1分之6,的前n项和

答案

6/(1*2)+6/(2*3)+6/(3*4)+.+6/[n(n+1)]
=6{1/(1*2)+1/(2*3)+6/(3*4)+.+1/[n(n+1)]}
=6*[1-1/2+1/2-1/3+1/3-1/4+.+1/n-1/(n+1)]
=6*[1-1/(n+1)]
=6n/(n+1)