Sigma summation notation to rewrite this sum, Check

Question

Can someone tell me if I did this correctly?

Given: (9/16)+(3/8)+(1/4)+(1/6)+(1/9)+(2/27)

I know that the common ratio is 2/3 and a_n=a₁r^n-1

So,

n

∑ i=(9*2^n-1)/(16*3^n-1)

i=9/16

Answer 1

the common term is C*r^n

n=1 ---> C*r = 9/16

n=2 ---> C*r^2 = 3/8

C *r * r = 3/8

9/16 * r = 3/8

9 r = 6

r = 6/9 = 2/3

C*(2/3) = 9/16

C = 9/16 * 3/2 = 27/32

The function is (27/32)(2/3)^n

The sum converges to 9/16 * 1 / (1 - 2/3) = 9/16 * 3 = 27/16