Chapter 3.5, Problem 21PPS

To determine

Write an equation for the nth terms of the sequence and then graph the first five terms in the sequence.

Answer to Problem 21PPS

  $a_n=0.25n-1$

Explanation of Solution

Given:

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

Concept Used:

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.

In an Arithmetic Sequence the difference between one term and the next is a constant.

Calculation:

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

Step1: Find the common difference.

Find the common difference:

  $a_1=-7a_2=-4a_3=-1$

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

The common difference is 0.25

Step 2: Write an equation.

  $\begin{array}{l}{a}_n=a_1+(n-1)d\text{Formula}\text{for}\text{the}nth\text{term}\\ a_n=-0.75+(n-1)·0.25a_1=-0.75\text{and}d=-0.25\\ a_n=-0.75+0.25n-0.25\text{Distributive}\text{Property}\\ a_n=0.25n-1\text{Simplify}.\end{array}$

The nth term of the sequence $-\mathrm{0..75},-0.50,-0.25,\mathrm{0....}$ is $0.25n-1$

Graph the five terms of the sequence:

n $0.25n-1$ $a_n$ $(n,a_n)$ 1 $0.25·1-1=-0.75$ − 0.75 (1, − 0.75) 2 $0.25·2-1=-0.50$ − 0.50 (2, − 0.50) 3 $0.25·3-1=-0.25$ − 0.25 (3, − 0.25) 4 $0.25·4-1=0$ 0 (4, 0) 5 $0.25·5-1=0.25$ 0.25 (5, 0.25)

Thus, the equation of nth terms of the sequence is $0.25n-1$