A problem in the recruitment of primary school mathematics teachers

( 1)、s 1 = 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

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)

Prove:

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)

……………………………………

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

Therefore: sn =1/2 * [1-1/3+1/5+1/5-65438+7+...+655.

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