Chapter 3.6, Problem 21PPS

To determine

To determine the equation and the graph of the given sequence.

Answer to Problem 21PPS

  $a_n=0.25n-1$

Explanation of Solution

Given:

Sequence:

  $-0.75,-0.5,-0.25,\mathrm{0...}$

Formula used:

nth term of any arithmetic progression is given by the formula:

  $t_n=a+(n-1)d$

  $t_n=$ nth term

  $a=$ The first term of the arithmetic progression

  $d=$ The common difference of the arithmetic progression

Calculation:

The common difference of the arithmetic progression is:

  $-0.5-(-0.75)=0.25$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=-0.75+(n-1)(0.25)$  $[\because a=-0.75,d=0.25]$

  $\begin{array}{l}\Rightarrow a_n=-0.75+0.25n-0.25\\ \Rightarrow a_n=0.25n-1\end{array}$

So, the nth term of the sequence is $0.25n-1.$

Calculation for graph:

  $(n,a_n)$ represent the points of the graph.

First four points are: $(1,-0.75),(2,-0.5),(3,-0.25),(4,0)$

To find 5^th point substitute n = 5 in nth term.

  $\begin{array}{c}{a}_n=0.25n-1\\ =0.25(5)-1\\ =1.25-1\\ =0.25\end{array}$

So, the fifth point: $(5,0.25)$

Graph:

  

Interpretation:

From the graph it can be observed that the horizontal extent of the graph is from $-\infty$ to $\infty$ .

And the vertical extent of the graph is also from $-\infty$ to $\infty$ .

So, both the domain and range of the function is $(-\infty ,\infty )$ .