Geometric Sequences formula

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 calculated using the formula: $$a_n = a_1 \times r^{(n-1)}$$ , where $$a_1$$ is the first term, $$r$$ is the common ratio, and $$n$$ is the term number.

The common ratio $$r$$ can be found by dividing any term by the previous term: $$r = \frac{a_n}{a_{n-1}}$$ .

The 5th term can be calculated as follows: $$a_5 = 3 \times 2^{(5-1)} = 3 \times 16 = 48$$ .

The common ratio is $$r = \frac{20}{5} = 4$$ .

The 3rd term is calculated as follows: $$a_3 = 2 \times 3^{(3-1)} = 2 \times 9 = 18$$ .

The sum of the first n terms of a geometric sequence is given by: $$S_n = a_1 \times \frac{1 - r^n}{1 - r}$$ , where @@r eq 1@@.

The 6th term is calculated as follows: $$a_6 = 1 \times \frac{1}{2}^{(6-1)} = 1 \times \frac{1}{32} = \frac{1}{32}$$ .

The 10th term is calculated as follows: $$a_{10} = 4 \times (-2)^{(10-1)} = 4 \times 512 = 2048$$ .

A sequence is geometric if the ratio of consecutive terms is constant.