I am creating a study sheet for geometric series and came across this questions
Do you think there is an infinite geometric series with first term 10 and a sum of 4? If so, find one infinite geometric series with first term 10 and a sum of 4, as well as explain how you get this infinite geometric series. If not, explain why.
This is possible with some neagitve common ratio, r, where $|r|<1$ correct?
You know that if $a$ is the first term of a geometric series and $r$ is the ratio, then the sum is given by $$S=\sum_{n=0}^\infty ar^n=\frac{a}{1-r}$$ So, if $a=10$ and $S=4$, then $$\frac{10}{1-r}=4\implies r=-\frac{3}{2}$$ but the series does not converge in this case, so the solution is extraneous and the described situation is not possible.