Geometric Sequences

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 common ratio is the factor by which we multiply each term to get the next term in the sequence.

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.

The 4th term is 3 * 2^(4-1) = 3 * 8 = 24.

The sum of the first n terms (S_n) of a geometric sequence is given by S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1.

The common ratio is 1/2.

The 14th term is 5 * 2^(14-1) = 5 * 8192 = 40960.

The next three terms are -324, -972, -2916.

a_5 = 5 * 2^(5-1) = 5 * 16 = 80.

The common ratio is -4.