Example: Sum of first 50 positive integers: $S_{50} = \frac{50}{2}(1 + 50) = 25 \cdot 51 = 1275$

Geometric Series

Finite:

$S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \quad (r \neq 1)$

Infinite (converges when $|r| < 1$):

$S = \frac{a_1}{1 - r}$

Example: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$ $S = \frac{1}{1 - \frac{1}{2}} = 2$

Sigma Notation

$\sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n$

Counting Principles

Fundamental Counting Principle

If there are $m$ ways to do one thing and $n$ ways to do another, there are $m \times n$ ways to do both.

Permutations (order matters)

$P(n, r) = \frac{n!}{(n-r)!}$

Combinations (order doesn't matter)

$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

Probability

$P(E) = \frac{\text{number of favorable outcomes}}{\text{total outcomes}}$

Addition Rule

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Multiplication Rule (Independent Events)

$P(A \cap B) = P(A) \cdot P(B)$

Binomial Probability

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Permutation vs Combination: Does the ORDER matter? If arranging things in a LINE → permutation. If choosing a GROUP → combination.