In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = n + 5

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. For example, in the sequence 2, 4, 6, 8, the common difference is 2, as each term increases by 2 from the previous term.

Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For instance, in the sequence 3, 6, 12, 24, the common ratio is 2, since each term is obtained by multiplying the previous term by 2.

Determining Sequence Type

To determine whether a sequence is arithmetic, geometric, or neither, one must analyze the relationship between consecutive terms. For an arithmetic sequence, check if the differences are constant; for a geometric sequence, check if the ratios of consecutive terms are constant. If neither condition holds, the sequence is classified as neither.