Q.Find Sn, the sum tonterms, of the geometric series 2 + 2/ 3 + 2/ 3^2 +…

If Sn = 242/ 81, find the value ofn.

Eq.Sn = a (1 - r^n) / (1 - r) … (r < 1).

Attempt:

From the series above, r = (2/ 3) / 2 => 1/ 3

Therefore, a = 2 and r = 1/ 3. Sub these into the Sn equation…

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

2 (1 - (1/ 3)^n) / (2 / 3)

Answer:

I’m stuck at this point, as the answer in the text book indicates that:

Sn = 3 - (1/ (3^(n-1)))

I am uncertain of the procedure they have applied to convert n into n - 1. Can anyone help? Thank you.