[s(n)]^2=an(s(n)- 1/2)=[s(n)-s(n- 1)][s(n)- 1/2]

[s(n)]^2=[s(n)-s(n- 1)]s(n)- 1/2[s(n)-s(n- 1]

[s(n)]^2=[s(n)]^2-s(n- 1)*s(n)- 1/2[s(n)-s(n- 1]

0 =-S(n- 1)* S(n)- 1/2[S(n)-S(n- 1)]

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

2S(n- 1)S(n)=-[S(n)-S(n- 1)]

2 = 1/ sec (n)- 1/ sec (n- 1)

1/S(n) is arithmetic progression.

S( 1)=a( 1)= 1

1/S(n)=2n- 1

S(n)= 1/(2n- 1)

a(n)= 1/(2n- 1)- 1/(2n-3)

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

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

TN =( 1/2)[( 1- 1/3)+( 1/3- 1/5)+……+ 1/(2n- 1)- 1/(2n+ 1)]

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

=n/(2n+ 1)