Chapter 4.1, Problem 71SR

To determine

To Calculate:

For given arithmetic sequence, determine the related function. Then determine if the function is proportional or non-proportional .

3,7,11…….

Answer to Problem 71SR

  $\begin{array}{l}{a}_n=4n-1\\ or\\ f(n)=4n-1\end{array}$

It is non-proportional.

Explanation of Solution

Concept Used:

1.

The graph of a non-proportional linear relationship is a line that does not cross through the origin, whereas the graph of a proportional linear relationship is a line that does cross through the origin.

2.

For A.P

  $a,a+d,a+2d,\mathrm{........}$

n-th term is given by:

  $\begin{array}{l}{a}_n=a+(n-1)d\\ where\\ a=first\_term\_of\_AP\\ d=common\_difference\end{array}$

3.

The slope-intercept form is given by :

  $\begin{array}{l}y=mx+b\\ where\\ m=slope=\frac{rise}{run}\\ b=y-\mathrm{int}ercept\end{array}$

Calculation:

Given AP:

3,7,11,…..

  $\begin{array}{l}a=3\\ d=7-3=11-7=4\\ nth-term:\\ a_n=a+(n-1)d\\ \Rightarrow a_n=3+(n-1)\mathrm{4...........}[a=3,d=4]\\ \Rightarrow a_n=3+4n-\mathrm{4.............}[simplify]\\ \Rightarrow a_n=4n-\mathrm{1...................}[simplify]\\ or\\ f(n)=4n-1\end{array}$

Graph:

y-intercept is -1 and slope is $\frac{4}{1}$

Plot (0,-1). Slope is $\frac{4}{1}$ , so rise 4 units from the point and then run 1 unit to the right. Plot the point (1,3). Joint the two points to get the line of the equation $f(n)=4n-1$ .

  

Since , the line does not contain the origin (0,0) , it is nonproportional.