Question

(1^2+2^2)/1*2+(2^2+3^2)/2*3+……+(100^2+101^2)/100*101=?

Answer 1

每个式子可以化成(n^2+（n+1)^2)/n*(n+1)
=(2n^2+2n+1)/n*(n+1)
=(2n*(n+1)+1)/n*(n+1)将式子拆开，可得：2+1/n*(n+1)
一共有100个式子所以原式=2*100+1/1*2+1/2*3+1/3*4+...+1/100*101
=200+1-1/2+1/2-1/3+...+1/100-1/101约
=200+1-1/101
=200+100/101

Answer 2

(1^2+2^2)/1*2+(2^2+3^2)/2*3+……+(100^2+101^2)/100*101
=[(2-1)²+4]/1*2+[(3-2)²+12]/2*3+…………+[(101-100)²+20200]/100*101
=[2+2+2+2+……+2]+[(1-1/2)+(1/2-1/3)+……+(1/100-1/101)]
=200+[(1+1/2+1/3+……+1/100)-(1/2+1/3+……+1/101)]
=200+(1-1/101)
=200+100/101