Arithmetic and Geometric Sequences Review

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 expressed as: $$a_n = a_1 + (n-1)d$$ , 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 expressed as: $$a_n = a_1 \times r^{(n-1)}$$ , where $$a_1$$ is the first term and $$r$$ is the common ratio.

The common difference $$d$$ can be found by subtracting any term from the subsequent term: $$d = a_{n+1} - a_n$$ .

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

A recursive formula defines each term of a sequence using the preceding term(s). For example, in an arithmetic sequence, $$a_n = a_{n-1} + d$$ .

The first term of the sequence is 100.

The first five terms are: 100, 20, 4, 4/5, 4/25.

In an arithmetic sequence, the difference between terms is constant (common difference), while in a geometric sequence, the ratio between terms is constant (common ratio).