A question in the recruitment of primary school mathematics teachers

(1), S1=1/2*(1-1/3)=1/3

S2=1/2*(1-1/3)+1 /2*(1/3-1/5)=1/2*(1-1/3+1/3-1/5)=1/2*4/5=2/5

S3=1/2*(1-1/3+1/3-1/5+1/5-1/7)=1/2*(1-1/7)=3/7

(2)Sn=1/2*[1-1/3+1/3-1/5+1/5-1/7+……+1/（2n-1）-1/(2n +1)]

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

Proof:

Because: 1/(1*3)=1/2*(1-1/3)

1/(3*5)=1/2*(1/3-1/5)

1/(5*7)=1/2*(1/5-1/7)

……………………………………………………< /p>

So: 1/[（2n-1）*1/(2n+1)]=1/2*[1/（2n-1）-1/(2n+1)]

So: Sn=1/2*[1-1/3+1/3-1/5+1/5-1/7+……+1/（2n-1）-1/(2n +1)]

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