Question

How many terms are there in a geometric series if the first term is 5, the common ratio is 3, and the sum of the series is 65?

Answer 1

$sn = \frac{a(1 - {r}^{n}) }{1 - r}$
hence,

65=5(1-3^n)/1-3
( multiply by -2 on both sides )
130=5(1-3^n)
26=1-3^n
27=3^n

from here, you can use lg to solve for n.

lg 27 = lg 3^n
lg 27 = n lg 3
n = lg 27 ÷ lg 3

hence, n = 3