sn= 1*( 1/3)^2+3*( 1/3)^3+5*( 1/3)^4+.....+(2n- 1)*( 1/3)^(n+ 1)

( 1/3)sn= 1*( 1/3)^3+3*( 1/3)^4+5*( 1/3)^5+.....+(2n- 1)*( 1/3)^(n+2)

Dislocation cancellation leads to:

(2/3)Sn

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

=( 1/9)+2{( 1/3)^3*[ 1-( 1/3)^(n- 1)]}/( 1-( 1/3))-(2n- 1)*( 1/3)^(n+2)

=( 1/9)+( 1/9)[ 1-( 1/3)^(n- 1)]-(2n- 1)*( 1/3)^(n+2)

So sn = {(1/3)+(1/3) [1-(1/3) (n-1)]-(2n-1) * (.

It is mainly the sum of dislocation elimination methods.

And the number of terms of the sum of intermediate geometric series is n- 1.

The rest is calculation and simplification.