Find the closed formula for $\sum_{i=0}^n a^{n-i} \cdot b^i$

Question

For $a, b, n \in \mathbb N^0$, find the closed formula for the sum of the following sequence: $$(a^{n} \cdot b^0),(a^{n-1} \cdot b^1),(a^{n-2} \cdot b^2) \ldots (a^2 \cdot b^{n-2}),(a^1 \cdot b^{n-1}),(a^0 \cdot b^n)$$ i.e. $$\sum_{i=0}^n a^{n-i} \cdot b^i.$$

Also, does this sum have a special name?

Answer 1

Hint: What do you get when you multiply this sum by $a-b$ ?

Answer 2

$$\sum_{i=0}^n a^{n-i} \cdot b^i=\frac{a^{n+1}-b^{n+1}}{a-b}$$

Answer 3

Hint: Divide the summation by $a^n$. The result should be much more familiar...