The sum of an infinite geometric series with first term a and common ratio r < 1 is given by $ S_n=a\cdot\dfrac{r^n-1}{r-1} $. The sum of a given infinite geometric series is $S_{\infty}=200 $ and the common ratio $r$ is 0.15. What is the second term $a_2$ of this series?
I'm confused on how to attack,can someone explain it to me? Thanks.
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$$s=a+aq+aq^2+aq^3+aq^4+...=200\\q=0.15\\$$ to find sum of terms multiply s by q , then find $s- qs$ $$s=a+aq+aq^2+aq^3+aq^4+...\\sq= aq+aq^2+aq^3+aq^4+aq^5...\\s-sq=a+(aq-aq)+(aq^2-aq^2)+...=a\\s(1-q)=a\\s=\frac{a}{1-q}\\200=\frac{a}{1-0.15}\\a=170\\$$ now you have a,q then the second term is $$aq=170(0.15)=25.5$$
Hint If $a$ is the first term and $|r|<1$ is the ratio you have $$a+ar+..+ar^{n}+....=a\frac{1}{1-r}$$
You are told that $a\frac1{1-r} =200$ and what $r$ is . Can you find $a$? Can you find the second term?