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Chapter 10: Problem 73

What is the common difference in an arithmetic sequence?

Short Answer

Expert verified
The common difference in an arithmetic sequence is the constant difference between each successive term in the sequence, calculated by subtracting a term from the term that directly follows it.

Step by step solution

01

Identify an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. For example, the sequence \(2, 4, 6, 8\) is an arithmetic sequence.
02

Compute the Common Difference

The common difference in an arithmetic sequence can be calculated by subtracting any term from the term that directly follows it. In the example given (2, 4, 6, 8), the common difference can be found by subtracting 2 from 4, which equals 2.
03

Verify the Common Difference

To verify that the computed difference is constant, subtract the first term from the second term, the second term from the third term, and so on. If the differences are equal, then you've correctly calculated the common difference. In our example, subtracting 2 from 4, 4 from 6, and 6 from 8 all result in 2, which confirms that the common difference is 2.

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Most popular questions from this chapter

This will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=3 \cdot 5^{n}\). Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

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Explain how to find the sum of the first \(n\) terms of an arithmetic sequence without having to add up all the terms.
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