Compute the sum $\sum_{i=0}^n 5^{i+1}-5^i$

Question

Compute the sum:

$$\sum_{i=0}^n 5^{i+1}-5^i$$

with the hint, "start by writing out (and expanding) the sum."

So I did and got

$$4 + 20 + 100...$$

with the appearance of going to infinity. Is infinity a valid answer?

Answer 1

The person who wrote the hint didn't quite get her meaning across. Expand it without simplifying: $$ \sum_{i=0}^n 5^{i+1}-5^i = (5^1-5^0) + (5^2-5^1) + (5^3-5^2) + \cdots + (5^{n+1}-5^n). $$ We see immediately that almost all of the terms cancel out. The sum is $5^{n+1}-1$. This is an example of a telescoping sum.

Answer 2

According to the hint $$\begin{align*}\sum_{i=0}^n (5^{i+1}-5^i)&=(5^1-5^0)+(5^2-5^1)+(5^3-5^2)+\ldots+(5^{n}-5^{n-1})+(5^{n+1}-5^{n})=\\&=-5^0+(5^1-5^1)+(5^2-5^2)+\ldots+(5^{n-1}-5^{n-1})+(5^{n}-5^{n})+5^{n+1}=\\\\&=5^{n+1}-5^0=5^{n+1}-1 \end{align*}$$