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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?

K Math
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1 Answers1

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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.

Franklin Pezzuti Dyer
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  • ok this is what I continued to get when I was solving but wasn't sure if I was missing something. – K Math Dec 04 '18 at 00:43