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Is there a way to sum up consecutive powers like this when we know the value of $a$ using a closed form expression or do we actually have to add up each one?

By closed form I mean in the same way that the sum of numbers $1$ to $n$ is $n(n+1)/2$.

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It's called a geometric sum.

$a + a^2 + \cdots + a^n = \frac{a - a^{n+1}}{1 - a}.$

If it's infinite, then we call it a geometric series and we have

$a + a^2 + \cdots = \frac{a}{1 - a}$

though this second one only converges for $|a| < 1$. The proof is similar to the formula you mention. Let $S = a + a^2 + \cdots + a^n$. Now, subtract $S - aS$.

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let $S_n=a+a^2+...+a^n$ then $aS_n=a^2+a^3+...+a^n+a^{n+1}$ from first equation subtract the second then $S_n-aS_n=a-a^{n+1}$ $S_n(1-a)=a-a^{n+1}$ $S_n=\frac{a-a^{n+1}}{1-a}$