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. a_n = n + 5

Verified step by step guidance

1. Identify the general term of the sequence: \(a_n = n + 5\).
2. Recall that an arithmetic sequence has a constant difference between consecutive terms, while a geometric sequence has a constant ratio between consecutive terms.
3. Calculate the first few terms of the sequence by substituting values of \(n\): for example, \(a_1 = 1 + 5\), \(a_2 = 2 + 5\), \(a_3 = 3 + 5\).
4. Find the differences between consecutive terms: \(a_2 - a_1\), \(a_3 - a_2\), and check if these differences are constant.
5. If the differences are constant, conclude the sequence is arithmetic and identify the common difference; if not, check the ratios \(\frac{a_2}{a_1}\), \(\frac{a_3}{a_2}\) to see if the sequence is geometric and find the common ratio if applicable.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference. For example, in the sequence 2, 5, 8, 11, the common difference is 3.

Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the sequence 3, 6, 12, 24, the common ratio is 2.

General Term of a Sequence

The general term (an) of a sequence is a formula that defines the nth term in terms of n. Understanding this formula helps identify the type of sequence and calculate specific terms, differences, or ratios as needed.