Chapter 3.6, Problem 6CYU

To determine

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

Answer to Problem 6CYU

  $a_n=0.5n-1.5$

Explanation of Solution

Given:

Sequence:

  $-1,-0.5,0,\mathrm{0.5...}$

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:

  $-0.5-(-1)=-0.5+1=0.5$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=-1+(n-1)(0.5)$  $[\because a=-1,d=0.5]$

  $\begin{array}{l}\Rightarrow a_n=-1+0.5n-0.5\\ \Rightarrow a_n=0.5n-1.5\end{array}$

So, the nth term of the sequence is $a_n=0.5n-1.5$ .

Calculation for graph:

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

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

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

  $\begin{array}{c}{a}_n=0.5n-1.5\\ =0.5(5)-1.5\\ =2.5-1.5\\ =1\end{array}$

So, the fifth point: $(5,1)$

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 $(-\infty ,\infty )$ .