Sequences and Series

A sequence is an ordered list of numbers that follow a specific pattern or rule.

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

The nth term (a_n) can be found using the formula: a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.

The nth term (a_n) can be found using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

The sum (S_n) can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where a_n is the nth term.

The sum (S_n) can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), for r ≠ 1.

The common difference (d) is the constant amount that each term increases or decreases by in an arithmetic sequence.

The common ratio (r) is the fixed number that each term is multiplied by to get the next term in a geometric sequence.

Geometric.