Chapter 3.5, Problem 36HP

To determine

Explain how to find a certain term of an arithmetic sequence and how an arithmetic sequence is related to a linear function.

Explanation of Solution

Concept Used:

We can find a term from the arithmetic sequence is:

Given an arithmetic sequence with the first term $a_1$ and the common difference d,

The nth (or general) term is given by $a_n=a_1+(n-1)d$

Example: The sequence: $-\mathrm{0..75},-0.50,-0.25,\mathrm{0....}$

  $a_1=-0.75a_2=-0.50a_3=-0.25$

  $d=a_2-a_1=-0.50-(-0.75)=-0.50+0.75=0.25\text{or}d=a_3-a_2=-0.25-(-0.50)=-0.25+0.50=0.25$

  $a_1=-0.75\text{and}d=0.25$

Fifth term of the sequence:

  $\begin{array}{l}{a}_n=a_1+(n-1)d\text{Formula}\text{for}\text{the}nth\text{term}\\ a_5=-0.75+(5-1)·0.25a_1=-0.75\text{and}d=-0.25\text{and}n=5\\ a_5=-0.75+4·0.25\\ a_5=-0.75+1=0.25\end{array}$

5^th term of the sequence $-\mathrm{0..75},-0.50,-0.25,\mathrm{0....}$ is 0.25

In this way we can find the specific term from a sequence.

Arithmetic sequence is a linear function:

An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence then subtract it by the previous one and the result is always the same or constant then it is an arithmetic sequence.

We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function.

They are similar in that the graph of the terms of an arithmetic sequence lines on a line. Therefore, an arithmetic sequence can be represented by a linear function. They are different in that the domain of an arithmetic sequence is the set of natural numbers, while the domain of a linear function is all real numbers.

Thus, arithmetic sequence is discrete while linear functions are continuous.

Consider the graph of the arithmetic sequence $a_n=4n-16$

  

For each n, there is a point on the graph.

The graph of the linear function: $y=4x-16$ is given below.

  

Thus, arithmetic sequence is discrete while linear functions are continuous.