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 (r) can be found by dividing any term by the previous term: r = a_n / a_(n-1).

The nth term of a geometric sequence can be expressed as: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

The 4th term is a_4 = 3 * 2^(4-1) = 3 * 8 = 24.

The common ratio is r = 15 / 5 = 3.

The next term is 54, found by multiplying the last term (18) by the common ratio (3).

The equation is a_n = 1 * 3^(n-1).

The common ratio is r = 50 / 100 = 0.5.

The sequence is arithmetic because the difference between consecutive terms is constant.

The terms will alternate in sign, as multiplying by a negative number changes the sign of the term.