Chapter 3.6, Problem 3CGP

To determine

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

Answer to Problem 3CGP

  $a_n=16-13n$

Explanation of Solution

Given:

Sequence:

  $3,-10,-23,-\mathrm{36...}$

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:

  $-10-3=-13$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=3+(n-1)(-13)$  $[\because a=3,d=-13]$

  $\begin{array}{l}\Rightarrow a_n=3-13n+13\\ \Rightarrow a_n=16-13n\end{array}$

So, the nth term of the sequence is $16-13n.$

Calculation for graph:

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

First four points are: $(1,3),(2,-10),(3,-23),(4,-36)$

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

  $\begin{array}{c}{a}_n=-13n+16\\ =-13(5)+16\\ =-65+16\\ =49\end{array}$

So, the fifth point: $(5,49)$

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 $(-\infty ,\infty )$ .