Help in understanding special algebra expansions

Question

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I cannot understand how this formula is called and in what cases it is applied. Can you help with this and give examples of equations using these formulas?

Answer 1

$$S = 1 + a + ... + a^{n-1}$$ $$aS = a + a^2 + ... + a^{n}$$ So by subtracting $$(a-1)S = a^n - 1$$ for $n \in \mathbb{N}$

So this is basically just geometric progression! You can derive the second result similarly - i.e. divide LHS and RHS by $(a+1)$ and observe that the sum $(1-a+a^2...-a^{2k-1}+a^{2k})$ is a geometric progression with common ratio $=-a$. Doing a manipulation similar to what I did above with $S$ will lead you to the result.

Examples of where these are used? Well, it's always nice to have some factorization tricks up your sleeve.