Calculating $e^{ar}+e^{ar^2}+..+e^{ar^n}$ Calculate the sum,

$$e^{ar}+e^{ar^2}+..+e^{ar^n}\ \text{where} \ a,r\in \mathbb{R}$$

It's easy to calculate the sum when the powers of $e$ are in A.P.
But, how to proceed when the powers are in G.P.?
Let ${ar}^i=i\log m.$ Then the series becomes $$\sum_{i=1}^{n}{e^{ar^{i}}}=\sum_{i=1}^{n}{e^{i\log{m}}}=\sum_{i=1}^{n}{{m}^i},$$ which is a geometric series.
Let ${ar}^i=i\log m.$ Then the series becomes $$\sum_{i=1}^{n}{e^{ar^{i}}}=\sum_{i=1}^{n}{e^{i\log{m}}}=\sum_{i=1}^{n}{{m}^i},$$ which is a geometric series.