Chapter 8.2, Problem 45E

To determine

To compare: The graph of $a_n=3n+1$ ,where n is a positive integer , with the graph $f(x)=3x+1$ , where x is a real number.

Answer to Problem 45E

The graphs are different. As $a_n$ represents the discrete points whereas $f(x)$ is the graph of continuous line.

Explanation of Solution

Given information:

  $a_{21}=-25,d=-\frac{3}{2}$ .

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d .

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

Calculation:

Consider,

  $a_n=3n+1$

$n$	$1$	$2$	$3$	$4$	$5$	$6$
$a_n$	$\begin{array}{l}{a}_n=3(1)+1\\ =3+1\\ =4\end{array}$	$\begin{array}{l}{a}_n=3(2)+1\\ =6+1\\ =7\end{array}$	$\begin{array}{l}{a}_n=3(3)+1\\ =9+1\\ =10\end{array}$	$\begin{array}{l}{a}_n=3(4)+1\\ =12+1\\ =13\end{array}$	$\begin{array}{l}{a}_n=3(5)+1\\ =15+1\\ =16\end{array}$	$\begin{array}{l}{a}_n=3(6)+1\\ =18+1\\ =19\end{array}$

Plotting the points we get,

  

Figure 1.

The x -axis is represented as $n$ and y -axis is represented as $a_n$ . The first six points of the sequence in the graph starting from the red dot to orange dot. The sequence is increasing.

Now, consider

  $f(x)=3x+1$

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$f(x)$	$\begin{array}{l}f(x)=3(1)+1\\ =3+1\\ =4\end{array}$	$\begin{array}{l}f(x)=3(2)+1\\ =6+1\\ =7\end{array}$	$\begin{array}{l}f(x)=3(3)+1\\ =9+1\\ =10\end{array}$	$\begin{array}{l}f(x)=3(4)+1\\ =12+1\\ =13\end{array}$	$\begin{array}{l}f(x)=3(5)+1\\ =15+1\\ =16\end{array}$	$\begin{array}{l}f(x)=3(6)+1\\ =18+1\\ =19\end{array}$

Plotting the points we get,

  

Figure 2.

The x -axis is represented as $x$ and y -axis is represented as $f(x)$ . The first six points of the sequence in the graph starting from the red dot to orange dot. The sequence is increasing.

From both the graphs we can conclude that $a_n$ is showing the discrete points whereas $f(x)$ is the graph of continuous line. Hence , both the graphs are different.