Question

设Sn=1/2+1/6+1/12+…+1/n（n+1）,且Sn*Sn+1=3/4,则n=多少?

Answer 1

Sn=1/2+1/6+1/12+…+1/n（n+1）

=1/(1*2)+1/(2*3)+...+1/(n)*(n+1)

=1-1/2+1/2-1/3+...+1/n-1/(n+1)

=1-1/(n+1)

=n/(n+1)

所以

Sn+1=1-1/(n+2)=(n+1)/(n+2)

sn*sn+1=3/4

n/(n+1)*(n+1)/(n+2)=3/4

n/(n+2)=3/4

4n=3n+6

n=6