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 = n² + 5

Verified step by step guidance

1. Identify the general term of the sequence: \(a_n = n^{2} + 5\).
2. Recall that an arithmetic sequence has a constant difference between consecutive terms, called the common difference \(d\), where \(d = a_{n+1} - a_n\).
3. Calculate the difference between consecutive terms: \(a_{n+1} - a_n = ((n+1)^2 + 5) - (n^2 + 5)\).
4. Simplify the expression for the difference: expand \((n+1)^2\) to get \(n^2 + 2n + 1\), then subtract \(n^2 + 5\) to find the difference.
5. Check if the difference is constant (independent of \(n\)). If it is, the sequence is arithmetic with common difference \(d\). If not, check if the ratio \(\frac{a_{n+1}}{a_n}\) is constant to determine if the sequence is geometric. If neither is constant, the sequence is neither arithmetic nor geometric.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list 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 fixed number called the common ratio. For example, in 3, 6, 12, 24, the common ratio is 2.

General Term of a Sequence

The general term, denoted a_n, defines the nth term of a sequence as a function of n. Understanding this formula helps determine the pattern of the sequence and whether it fits arithmetic or geometric rules.