Question

1+1/3+1/6+1/10+1/15+…+1/5050=?

求高手指点!

Answer 1

原式＝1+1/(1+2)+...+1/(1+2+3+...+n)

根据等差数列求和公式:1+2+3+...+n = n*(n+1)/2
1/(1+2)=1/[（1+2）×2÷2]=2/2×3=2×(1/2-1/3)
1/(1+2+3)=1/[（1+3）×3÷2]=2/3×4=2×(1/3-1/4)
......
1/(1+2+3+...+n)=1/[（1+n）×n÷2]=2/n×（n+1）=2×[1/n-1/(n+1)]
原式=1+2×(1/2-1/3+1/3-1/4......+1/n-1/(n+1)
=1+1-2/(n+1)
=2n/(n+1)

原式中5050＝1+2+...+100 所以n＝100
原式＝200/101

Answer 2

1+1/3+1/6+1/10+1/15+…+1/5050
=2/1*2+2/2*3+2*/3*4+2/4*5+2/5*6+…+2/100*101
=2*(1-1/2+1/2-1/3+1/3-1/4+…+1/100-1/101)
=2*(1-1/101)
=200/101

Answer 3

裂项法
1+1/3+1/6+1/10+...+1/5050
=1+2/6+2/12+2/20+…+2/10100
=1+(2/2×3)+(2/3×4)+(2/4×5)+…+(2/100×101)
=1+2(1/2-1/3+1/3-1/4+1/4-1/5+…+1/100-1/101）
=1+2(1/2-1/101)
=200/101.

Answer 4

利用节节相消法

1/(1+2+...+n) = 2/(n)(n+1) = 2(1/n - 1/(n+1))

1+1/3+1/6+1/10+1/15+。。。。。+1/1+2+3+。。。+2004
=2 (1/1-1/2+1/2-1/3+1/3-1/4 + ...-1/2005)
=2 (1-1/2005)
=4008/2005