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 = 2^n

Verified step by step guidance

1. <Start by understanding the general term of the sequence, which is given as a_n = 2^n.>
2. <To determine if the sequence is arithmetic, check if the difference between consecutive terms is constant. Calculate a_2 - a_1, a_3 - a_2, and so on.>
3. <To determine if the sequence is geometric, check if the ratio between consecutive terms is constant. Calculate a_2 / a_1, a_3 / a_2, and so on.>
4. <If the difference is constant, the sequence is arithmetic, and the constant difference is the common difference.>
5. <If the ratio is constant, the sequence is geometric, and the constant ratio is the common ratio.>

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. To determine if a sequence is arithmetic, one can subtract each term from the subsequent term and check for consistency.

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. To identify a geometric sequence, one can divide each term by its preceding term and verify if the ratio remains constant.

Exponential Functions

Exponential functions are mathematical functions of the form f(n) = a * b^n, where 'a' is a constant, 'b' is the base, and 'n' is the exponent. In the given sequence an = 2^n, it represents an exponential growth pattern where each term is a power of 2. Understanding exponential functions is crucial for analyzing sequences that grow or decay at rates proportional to their current value.