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Chapter 5: Problem 136

What is an arithmetic sequence? Give an example with your description.

Short Answer

Expert verified
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference. For example, in the sequence 2, 4, 6, 8, 10, the common difference is 2 as each term increases by 2 from the previous term.

Step by step solution

01

Definition of an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always constant. This difference is known as the common difference, and it can be positive, negative, or zero.
02

Illustrating through an example

Consider the sequence 2, 4, 6, 8, 10. Now, identify the difference between two consecutive terms. From 2 to 4, the difference is 2. The same goes for 4 to 6, 6 to 8, and 8 to 10, which means the common difference is 2. Therefore, this sequence is an arithmetic sequence.
03

Arithmetic Sequence Formula

The formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1) * d\) where \(a_n\) is the nth term, \(a_1\) is the first term, n is the term number and d is the common difference. In our example, for the 3rd term, \(a_3 = 2 + (3 - 1) * 2 = 6\), which matches our series.
04

Graphical Representation

If we plot this sequence on a number line or a graph, we will notice that each point is equally spaced. This is a key feature of an arithmetic sequence which visually pinpoints the concept of a 'common difference'.

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=-3\)

Use the appropriate formula shown above to find \(2+4+6+8+\cdots+200\), the sum of the first 100 positive even integers.

For the first 30 days of a flu outbreak, the number of students on your campus who become ill is increasing. Which is worse: The number of students with the flu is increasing arithmetically or is increasing geometrically? Explain your answer.

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{100}\), when \(a_{1}=50, r=1\).

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(15,30,60,120, \ldots\)
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