Chapter 3.6, Problem 18PPS

To determine

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

Answer to Problem 18PPS

  $a_n=-5n+2$

Explanation of Solution

Given:

Sequence:

  $-3,-8,-13,-\mathrm{18...}$

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:

  $-8-(-3)=-8+3=-5$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=-3+(n-1)(-5)$  $[\because a=-3,d=-5]$

  $\begin{array}{l}\Rightarrow a_n=-3-5n+5\\ \Rightarrow a_n=2-5n\end{array}$

So, the nth term of the sequence is $a_n=-5n+2$ .

Calculation for graph:

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

First four points: $(1,-3),(2,-8),(3,-13),(4,-18)$

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

  $\begin{array}{c}{a}_n=-5n+2\\ =-5(5)+2\\ =-25+2\\ =-23\end{array}$

So, the fifth point: $(5,-23)$

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 $(-\infty ,\infty )$ .