Question

若已知:1乘2分之1=1-2分之一, 2乘3分之1=2分之1-3分之1, 3乘4分之1=3分之1-4分之1...........

(1)计算1乘2分之1+2乘3分之1+3乘4分之1+......+2003乘2004分之1

(2)计算1乘2分之1+2乘3分之1+3乘4分之1+......+nx(n+1)分之1

虽然描述不太清楚,但勉强能看明白,求各位大虾帮我解最后一道题.

Answer 1

1×1/2+2×1/3+3×1/4+......+2003×1/2004
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+......+(1/2003-1/2004)
=1-1/2+1/2-1/3+1/3-1/4+1/4+......-1/2003+1/2003-1/2004
=1-1/2004
=2003/2004

1×1/2+2×1/3+3×1/4+......+n×1/(n+1)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+......+[(1/n)-(1/(n+1))]
=1-1/2+1/2-1/3+1/3-......-1/n+1/n-1/(n+1)
=1-1/(n+1)
=n/(n+1)