Is there a formula to get the answer of $a^1+a^2+a^3+\cdots+a^k\pmod{10^9+7}$?

Question

Hi so I am working on a problem where I have to calculate $\sum\limits_{i=1}^n a^i$ modulo $10^9+7$.

I have to do this on a computer, so I have to avoid overflow. I was wondering if there will be a formula that can help me with my problem.

Answer 1

in general, if $a\neq 1$ $$\sum_{k=0}^n a^k = \frac{a^{n+1} -1}{a-1}$$ For a proof, multiply by $a-1$ and observe the "telescopic" behavior of the lhs.

(Note that the sum starts with $k=0$, not $k=1$ as in your case).