I am learning about geometric sequences and I came across the question "For any number $x>2$, show that there is an infinite geometric series $a_n$ such that $a_0=2$ and $\sum_{n=0}^{\infty}a_n=x$."
Geometric Progressions
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sequences-and-series
geometric-series
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1What is this $a?$ – 2017-01-31
1 Answers
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The sum of an infinite geometric series with first term $a$ and ratio $r$ is $\frac a{1-r}$ as long as $|r| \lt 1.$ Here we are given $a=2$, so you just need to show that if you want a sum $x \gt 2$ you can solve $x=\frac 2{1-r}$ with $|r| \lt 1$. In fact you can get any positive number, not just those greater than $2$, but the question does not ask that.