Calculus II - Infinite Series - Geometric Series

[GreenPrint]

Homework Statement

Hi,

I'm trying to solve the problem in the attachment. I was asked to evaluate the left hand side equation of the equal sign. I was unsure how to go about evaluating it so I consulted my solutions manual to look up the first step. The right hand side equation of the equal sign is what my solutions manual did for the first step. I do not see how the two are equal and what intermediate steps were left out to prove that the two are equal. I was hoping someone could explain to me what was done. I have the feeling that whatever intermediate steps were performed to go from the right hand side to the left hand side of the equal sign are very simple and is the reason why they were left hand but I can't seem to figure it out. Thanks for any help!

Homework Equations

The Attempt at a Solution

 

[SammyS]

5^6-k=5⁶/5^k

 

[kru_]

$$\sum\left(\frac{1}{4}\right)^k 5^{6-k} = \sum\left(\frac{1^k}{4^k}\right)\frac{5^6}{5^k} = \sum\frac{5^6}{4^k 5^k} = 5^6 \sum\frac{1}{\left(4 * 5\right)^k}= 5^6 \sum\frac{1}{20^k} = 5^6\sum\left(\frac{1}{20}\right)^k$$

 

[GreenPrint]

Lol an algebra II thing with exponents >_>, much thanks