Question

1除以1乘2乘3,加上1除以2乘3乘4,加上1除以3乘4乘5,一直加到1除以20乘21乘22,等于多少

Answer 1

1/[n*(n+1)*(n+2)]

=[1/n-1/(n+1)]/(n+2)

=[1/n-1/(n+2)]/2-[1/(n+1)-1/(n+2)]

=[1/n-2/(n+1)+1/(n+2)]/2

1/(1*2*3)=[1/1-2/2+1/3]/2

1/(2*3*4)=[1/2-2/3+1/4]/2

.

1/(20*21*22)=1/[1/20-2/21+1/22]/2

1/(2*3*4)+1/(3*4*5)+...+1/(20*21*22)

=[1/1-2/2]/2+[1/2]/2+[1/21]/2+[-2/21+1/22]/2

=1/4+1/42-1/21+1/11

=