1乘2乘3分之1加2乘3乘4分之1加3乘4乘5分之1……加48乘49乘50分之1

题目

1乘2乘3分之1加2乘3乘4分之1加3乘4乘5分之1……加48乘49乘50分之1

答案

1/(1*2*3)+1/(2*3*4)
an=1/[n(n+1)(n+2)]=[(n+2)-(n+1)]/[n(n+1)(n+2)]
=1/n(n+1)-1/n(n+2)
=[1/n-1/(n+1)]-(1/2)[1/n-1/(n+2)]
a1+a2+..+a48
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+..+(1/48-1/49)
-(1/2)[ (1-1/3)+(1/2-1/4)+(1/3-1/5)+..+(1/47-1/49)+(1/48-1/50)]
=(1-1/49) -(1/2)[(1-1/49)+(1/2-1/50)]
=(1/2)(1-1/49)-(1/2)(1/2-1/50)
=1/2-1/98-1/4+1/100
=1/4-1/98+1/100