Question

1/1*2*3+1/2*3*4+1/3*4*5+......+1/99*100*101=?简便方法 帮帮忙

Answer 1

1/(n*(n+1)(n+2))=﹛1/[n*(n+1)]-1/[(n+1)(n+2)]﹜/2=[1/n-2/(n+1)+1/(n+2)]/2
1/1*2*3+1/2*3*4+1/3*4*5+......+1/99*100*101
=1/2*(1-2*1/2+1/3+1/2-2*1/3+1/4+1/3-2*1/4+1/5+……+1/100+1/99-2*1/100+1/101)
=1/2*(1-1/2-1/100+1/101)
=5049/20200

追问

1+2²/2²-1+3²/3²-1+4²/4²-1+...+100²/100²-1=（）

追答

1
22/22=1
22/22-1=0 32/32-1=0 ……1002/1002-1=0
1+22/22-1+32/32-1+42/42-1+...+1002/1002-1
=1+(22/22-1)+(32/32-1)+……+(1002/1002-1)
=1+0
1