Sequences & Series

A sequence is a list that follows a rule, and a series adds one up. These articles cover sequences that add a fixed amount and ones that multiply by a fixed ratio, how an infinite series can have a finite sum, sigma notation, and Pascal's triangle with the binomial theorem.

Arithmetic Sequences: The nth Term and Gauss's Sum Formula
An arithmetic sequence adds the same common difference $d$ every step, like $5, 8, 11, 14, \dots$ Its $n$ th term is $a_n = a_1 + (n - 1)d$ , and the sum of its first $n$ terms is $\frac{n}{2}(a_1 + a_n)$ .
Geometric Sequences and Infinite Geometric Series
A geometric sequence multiplies by the same common ratio $r$ every step, like $3, 6, 12, 24, \dots$ Its $n$ th term is $a_1 r^{n-1}$ . When $|r| < 1$ , adding all of its infinitely many terms gives a finite total, $\frac{a_1}{1 - r}$ .
Sigma Notation (Σ) Explained: How to Read and Write Summations
Sigma notation uses the Greek capital letter $\Sigma$ to mean add up. $\sum_{k=1}^{4} (2k + 1)$ means substitute $k = 1, 2, 3, 4$ into $2k + 1$ and add the results.
The Binomial Theorem and Pascal's Triangle
The binomial theorem expands $(a + b)^n$ directly. The coefficients come from row $n$ of Pascal's triangle, or from the formula $\binom{n}{k} = \frac{n!}{k!(n - k)!}$ .