Arithmetic and Geometric Sequences

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

The nth term of an arithmetic sequence 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 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 of a geometric sequence 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.

a_12 = 5 + (12-1) * 3 = 5 + 33 = 38.

a_12 = 2 * 3^(12-1) = 2 * 3^11 = 2 * 177147 = 354294.

The common difference d = 7 - 10 = -3.

The common ratio r = 9 / 3 = 3.

Next three terms: 11, 14, 17.

Next three terms: 108, 324, 972.