Chapter 3.6, Problem 20PPS

To determine

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

Answer to Problem 20PPS

  $a_n=-4n-7$

Explanation of Solution

Given:

Sequence:

  $-11,-15,-19,-\mathrm{23...}$

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:

  $-15-(-11)=-15+11=-4$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=-11+(n-1)(-4)$  $[\because a=-11,d=-4]$

  $\begin{array}{l}\Rightarrow a_n=-11-4n+4\\ \Rightarrow a_n=-4n-7\end{array}$

So, the nth term of the sequence is $-4n-7$ .

Calculation for graph:

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

First four points are: $(1,-11),(2,-15),(3,-19),(4,-23)$

To find fifth point, substitute n = 5 in nth term formula.

  $\begin{array}{c}{a}_n=-4n-7\\ =-4(5)-7\\ =-20-7\\ =-27\end{array}$

So, the fifth point is $(5,-27).$

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 $(-\infty ,\infty )$ .