Sequences and Series

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

A series is the sum of the terms of a sequence.

The common difference is the constant amount that each term in an arithmetic sequence differs from the previous term.

The nth term 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.

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

The common ratio is the constant factor between consecutive terms in a geometric sequence.

The nth term 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.

An infinite geometric series is the sum of the terms of a geometric sequence that continues indefinitely.

An infinite geometric series converges if the absolute value of the common ratio is less than 1 (|r| < 1).

The sum S of an infinite geometric series can be calculated using the formula: S = a / (1 - r), where a is the first term and r is the common ratio.