$\displaystyle \sum_{k=1}^n a^k b^k=?$
Hello, jut!
Simplify: .$\displaystyle \sum_{k=1}^n a^k b^k$
We have: .$\displaystyle S \;=\;ab + a^2b^2 + a^3b^3 + \hdots + a^nb^n$
This is a geometric series with: .$\displaystyle \begin{Bmatrix}\text{first term: }ab \\ \text{common ratio: }ab \\ n\text{ terms} \end{Bmatrix}$
Its sum is: .$\displaystyle S \;=\;ab\,\frac{(ab)^n - 1}{ab-1} $