Question: Find $\displaystyle \sum \limits_{n=1}^{\infty} \frac{2^{2n+1}}{5^n}$
My issue(s): How do I do this without using a calculator? I know that I have to do something with $S_n$ and $S_{n+1}$ but I'm not sure what.
Question: Find $\displaystyle \sum \limits_{n=1}^{\infty} \frac{2^{2n+1}}{5^n}$
My issue(s): How do I do this without using a calculator? I know that I have to do something with $S_n$ and $S_{n+1}$ but I'm not sure what.
Use properties of exponents to rewrite $\frac{2^{2n+1}}{5^n}$ in the form $A(B)^n.$ You probably recognize that form as a geometric sequence/series. The sum of an infinite geometric series with first term $a_1$ and constant ratio $r$ with $-1
HINT Try to write this as a geometric series by pulling out a factor of $2$ and rewriting $2^{2n}$ as $4^n$
Let $\displaystyle{S=\sum_{n=1}^{\infty} \frac{2^{2n+1}}{5^n}=\sum_{n=1}^{\infty} 2(\frac{4}{5})^n}$.
Then, $S=2(\frac{4}{5}+\frac{16}{25}+\frac{64}{125}+...)$
And, $\frac{4S}{5}=2(\frac{16}{25}+\frac{64}{125}+\frac{256}{625}+...)$
Now, what happens if you subtract the second equation from the first?