Question

1+1/(1+2)+1/(1+2+3)+…+1/(1+2+3+4+…+50)=小学解法

Answer 1

找规律：
1 = 2/(1×2) = 2×(1 - 1/2)
1/(1+2) =1/3=2/6= 2/(2×3) = 2×(1/2 - 1/3)
1/(1+2+3) =1/6=2/12= 2/(3×4) = 2×(1/3 - 1/4)
1/(1+2+3+4) =1/10=2/20= 2/(4×5) = 2×(1/4 - 1/5)
·
·
·
1/(1+2+3+4+…+50)=1/1275=2/2550=2/(50×51) = 2×(1/50 - 1/51)
所以，
1+1/(1+2)+1/(1+2+3)+…+1/(1+2+3+4+…+50)
= 2×(1 - 1/2) + 2×(1/2 - 1/3) + 2×(1/3 - 1/4) + 2×(1/4 - 1/5) + …… + 2×(1/50 - 1/51)
= 2×(1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + …… + 1/50 - 1/51)
= 2×(1 - 1/51)
=100/51