Looking for the Closed Form of a Two-Variable Geometric Sum

Question

Is there any closed form of the equation $$\sum_{i=0}^n a^{n-i} \cdot b^i$$ for real values $a$ and $b$ and integer $n \ge 0$?

Answer 1

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

Answer 2

Yes, of course: for $a\ne b$, it's $$\frac{a^{n+1}-b^{n+1}}{a-b}$$

Answer 3

hint: Write it as $$a^n\left(\frac{b}{a}\right)^i$$