I need to clarify if the way of working out is correct for finding the common ratio, given the first term of a geometric series and its sum to infinity.
First term: a = 8
Given Sum: S = 400
Ratio: r = ?
My procedure is:
S = 400 = 8 $\frac{1-r∞}{ 1 - r}$
S = 400 = $\frac{8}{1 - r}$
400 $\cdot$ $(1 - r)$ = 8
(1 - r) = $\frac{8}{400}$
-r = $\frac{8}{400}$ - 1
r = -1 ($\frac{8}{400}$ -1)
r = 0.98
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]