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Chapter 11: 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 value obtained when we subtract any term in the sequence (except the first one) from its immediately preceding term.

Step by step solution

01

Understanding Arithmetic Sequence

In mathematics, an arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which the difference of any two successive members is a constant. This means each term in the sequence after the first is created by adding a certain value to the previous term.
02

Identifying the Common Difference

The constant difference that you get after subtracting any term in the sequence (except the first one) from its immediately preceding term is called the 'common difference'. It can be positive, negative, or zero.
03

Calculation of Common Difference

To calculate the common difference in an arithmetic sequence, subtract the first term from the second term. Alternately, you can subtract any term in the sequence from the term that immediately follows it. The result is the common difference. Symbolically, if an arithmetic sequence has terms denoted as \( a_1, a_2, a_3, a_4, \ldots, a_n \), then common difference 'd' can be calculated as \( d = a_2 - a_1 \) or \( d = a_{n+1} - a_n \).

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Most popular questions from this chapter

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?

A new factory in a small town has an annual payroll of S6 million. It is expected that \(60 \%\) of this money will be spent in the town by factory personnel. The people in the town who receive this money are expected to spend \(60 \%\) of what they receive in the town, and so on. What is the total of all this spending, called the total economic impact of the factory, on the town each year?

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\). A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. $$ \begin{array}{cccc} {16,} & {0.96(16),} & {(0.96)^{2}(16),} & {(0.96)^{3}(16)} \\ {\text { Ist }} & {2 \text { nd }} & {3 \text { rd }} & {4 \text { th }} \\ {\text { swing }} & {\text { swing }} & {\text { swing }} & {\text { swing }} \end{array} $$ After 10 swings, what is the total length of the distance the pendulum has swung?

Explaining the Concepts Explain how to find the probability of an event not occurring. Give an example.

In the sequence \(21,700,23,172,24,644,26,116, \ldots,\) which term is \(314,628 ?\)
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