Common Algebraic Patterns in CP

Here are algebraic formulas you'll encounter repeatedly in competitive programming:

$\bullet$ Sum of first $n$ integers: $1 + 2 + ... + n = \frac{n(n+1)}{2}$

$\bullet$ Sum of first $n$ squares: $1^2 + 2^2 + ... + n^2 = \frac{n(n+1)(2n+1)}{6}$

$\bullet$ Geometric series: $1 + r + r^2 + ... + r^{n-1} = \frac{r^n - 1}{r - 1}$ for $r \neq 1$

$\bullet$ Number of pairs: $\binom{n}{2} = \frac{n(n-1)}{2}$