Binomial summation without coefficient

Question

I have the following summation:

$$\sum_{i=0}^na^{n-i}b^i$$

I recognise that if the binomial coefficient was present, it would represent the expansion of $(a+b)^n$. However, since that coefficient is absent, I am struggling to find a formulaic representation of the summation: does one exist?

Answer 1

\begin{align} \frac{a^2-b^2}{a-b}\quad &=a+b,\\ \frac{a^3-b^3}{a-b}\quad &=a^2+ab+b^2,\\ \ldots & \\ \frac{a^{n+1}-b^{n+1}}{a-b}&=\; ?\; \end{align}