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. a_n = n² + 5

Verified step by step guidance

1. Recall the definitions: An arithmetic sequence has a constant difference between consecutive terms, called the common difference (d). A geometric sequence has a constant ratio between consecutive terms, called the common ratio (r).
2. Write the general term of the sequence as $a{n}_{}=n^2+5$.
3. Calculate the first few terms by substituting values of $n$: $a\_1\; =\; 1^2\; +\; 5\; =\; 6$, $a\_2\; =\; 2^2\; +\; 5\; =\; 9$, $a\_3\; =\; 3^2\; +\; 5\; =\; 14$, and so on.
4. Find the differences between consecutive terms: $a\_2\; -\; a\_1\; =\; 9\; -\; 6\; =\; 3$, $a\_3\; -\; a\_2\; =\; 14\; -\; 9\; =\; 5$. Since these differences are not constant, the sequence is not arithmetic.
5. Find the ratios between consecutive terms: $\backslash frac\{a\_2\}\{a\_1\}\; =\; \backslash frac\{9\}\{6\}\; =\; 1.5$, $\backslash frac\{a\_3\}\{a\_2\}\; =\; \backslash frac\{14\}\{9\}\; \backslash approx\; 1.56$. Since these ratios are not constant, the sequence is not geometric.

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. It helps identify the pattern of the sequence and determine if it is arithmetic, geometric, or neither by analyzing how terms change as n increases.