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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 53
Chapter 9, Problem 53

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 = 2n

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Identify the general term of the sequence, which is given as an = 2n.
Recall the definitions: an arithmetic sequence has a constant difference between consecutive terms, and a geometric sequence has a constant ratio between consecutive terms.
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.
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.
If the differences are not constant, check the ratio of consecutive terms: compute \(\frac{a_2}{a_1}\) and \(\frac{a_3}{a_2}\). If these ratios are equal, the sequence is geometric and the common ratio is that value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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