Determine the sum of the following series
(4^n+8^n)/(13^n)
I know the answer should be 92/45 but how do I attain that? Determine the sum of the following series
(4^n+8^n)/(13^n)
I know the answer should be 92/45 but how do I attain that? @Mathematics

Question

anonymous · Answer

One approach is to algebraically rewrite the expression to$$\frac{4^n+8^n}{13^n}=\frac{4^n}{13^n}+\frac{8^n}{13^n}=\left( \frac{4}{13} \right)^n+\left( \frac{8}{13} \right)^n$$Then you have two infinite geometric series that you can sum separately, then add using the formula$$s=\frac{a _{1}}{1-r}$$where a1 is the first term and r is the common ratio. Thus, you have$$\frac{4/13}{1-4/13}+\frac{8/13}{1-8/13}=\frac{4}{9}+\frac{8}{5}=\frac{92}{45}$$ The other approach is partial fractions (see link): http://www.wolframalpha.com/input/?_=1321481252394&i=sum+%284^n%2b8^n%29%2f13^n%2c+n%3d1+to+infinity&fp=1&incTime=true