Chapter 3.6, Problem 19PPS

To determine

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

Answer to Problem 19PPS

  $a_n=5n-7$

Explanation of Solution

Given:

Sequence:

  $-2,3,8,\mathrm{13...}$

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:

  $3-(-2)=3+2=5$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=-2+(n-1)(5)$  $[\because a=-2,d=5]$

  $\begin{array}{l}\Rightarrow a_n=-2+5n-5\\ \Rightarrow a_n=5n-7\end{array}$

So, the nth term of the sequence is $5n-7$ .

Calculation for graph:

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

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

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

  $\begin{array}{c}{a}_n=5n-7\\ =5(5)-7\\ =25-7\\ =18\end{array}$

So, the fifth point: $(5,18)$

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 $(-\infty ,\infty )$ .