Question

1+ 1+2分之1+ 1+2+3分之1+……+1+2+3+……+100分之一等于多少？

用裂项相消法解决

Answer 1

原式=1+2/(2×3)+2/(3×4)+...+2/(100×101)=1+2(1/2-1/3+1/3-1/4+1/100-1/101)
=1+2(1/2-1/101)
=1+2×99/202
=1+99/101
=200/201

Answer 2

=2/2 +2/2×3 +2/3×4 +……+2/100×101
=2×[1/1×2+1/2×3+1/3×4+……+1/100×101]
=2×(1-1/2+1/2-1/3+1/3-1/4+……+1/100-1/101)
=2×100/101
=200/101