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 = 2ⁿ

Verified step by step guidance

1. Identify the general term of the sequence, which is given as $a{n}_{}=2^n$.
2. Recall the definitions: an arithmetic sequence has a constant difference between consecutive terms, and 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, $a\_1\; =\; 2^1\; =\; 2$, $a\_2\; =\; 2^2\; =\; 4$, $a\_3\; =\; 2^3\; =\; 8$.
4. Check if the difference between consecutive terms is constant: compute $a\_2\; -\; a\_1$ and $a\_3\; -\; a\_2$. If these are equal, the sequence is arithmetic.
5. If the differences are not constant, check the ratio of consecutive terms: compute $\backslash frac\{a\_2\}\{a\_1\}$ and $\backslash frac\{a\_3\}\{a\_2\}$. If these ratios are equal, the sequence is geometric and the common ratio is that value.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant is called the common difference. For example, in the sequence 3, 5, 7, 9, the common difference is 2.

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 2, 4, 8, 16, the common ratio is 2.

General Term of a Sequence

The general term (an) of a sequence defines the nth term as a function of n. Understanding this formula helps identify the type of sequence and calculate specific terms, differences, or ratios. For example, an = 2^n defines each term as 2 raised to the power n.