Sum of potencies with higher potency as clue

Question

I am supposed to calculate the following as simple as possible.

Calcute: $$1 + 101 + 101^2 + 101^3 + 101^4 + 101^5 + 101^6 + 101^7$$ Tip: $$ 101^8 = 10828567056280801$$

I have absolutely no idea how this tip is supposed to help me.
Do I still have to calculate each potency?
Can I somehow solve it with 101^7 * (1 + 101) = 10828567056280801?

As I am not allowed to use a calculator a simple technique for formulas like the one above would be welcome.

Answer 1

Hint: $$1 + x + x^2 + \dots + x^n = \frac{x^{n+1} - 1}{x - 1}.$$

Answer 2

Hint: Let $S=1 + 101 + 101^2 + 101^3 + 101^4 + 101^5 + 101^6 + 101^7$ and consider $101S$.