I need to clarify if the way of working out is correct in finding the common ration by knowing the first term of a geometric series and the sum to infinity.
first term: a = 8
sum: S = 400
ratio: r = ?
my workings are:
S400 = 8 (1-r∞)/ 1 – r
S400 = 8 / 1 – r
400 * (1 – r) = 8
(1 – r) = 8/400
-r = (8/400) – 1
r = -1 ((8/400)-1)
r = 0.98
Best Answer
Your reasoning is correct, but please don't write things like $r^\infty$, this is not well defined. You are encouraged to start your solution like this:
Given a geometric sequence $a_{n+1}=r\cdot a_n$ for $n\in\mathbb{N}_0$ with $a_0=8$, $|r|<1$ and the sum $$ S=\sum_{n=0}^\infty a_n = \frac{a_0}{1-r}$$ given by $S=400$, we can calculate $r$ as such: $400=\frac{8}{1-r}$ [your steps following]