Question

Identify the sequence graphed below and the average rate of change from n = 1 to n = 3. coordinate plane showing the point 1, 8, point 2, 4, point 4, 1, and point 5, .5

Answer 1

Answer:

Sequence : $a_n=8(\frac{1}{2})^{n-1}$

Rate of change : -3

Step-by-step explanation:

It is given that the sequence represents by the points (1,8), (2,4), (4,1) and (5,0.5).

We know that the $a_n=k$ represent the point (n,k). So, we have

$a_1=8,a_2=4,a_4=1,a_5=0.5$

$\frac{a_2}{a_1}=\frac{4}{8}=\frac{1}{2}$

$\frac{a_5}{a_4}=\frac{0.5}{1}=\frac{1}{2}$

It is clear that the above sequence is a geometric sequence because it has common ratio $\frac{1}{2}$.

First term : $a=8$

Common ratio : $r=\frac{1}{2}$

The nth term of a G.P. is

$a_n=ar^{n-1}$

$a_n=8(\frac{1}{2})^{n-1}$

Therefore, the required sequence is $a_n=8(\frac{1}{2})^{n-1}$.

We need to find the average rate of change from n = 1 to n = 3.

$a_3=8(\frac{1}{2})^{3-1}=2$

$Slope=\dfrac{a_3-a_1}{3-1}$

$Slope=\dfrac{2-8}{2}$

$Slope=\dfrac{-6}{2}$

$Slope=-3$

Therefore, the average rate of change from n = 1 to n = 3 is -3.