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

Sequence :
`a_n=8((1)/(2))^(n-1)`

Rate of change : -3

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`

`(a_2)/(a_1)=(4)/(8)=(1)/(2)`

`(a_5)/(a_4)=(0.5)/(1)=(1)/(2)`

It is clear that the above sequence is a geometric sequence because it has common ratio
`(1)/(2)`.

First term :
`a=8`

Common ratio :
`r=(1)/(2)`

The nth term of a G.P. is

`a_n=ar^(n-1)`

`a_n=8((1)/(2))^(n-1)`

Therefore, the required sequence is
`a_n=8((1)/(2))^(n-1)`.

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

`a_3=8((1)/(2))^(3-1)=2`

`Slope=(a_3-a_1)/(3-1)`

`Slope=(2-8)/(2)`

`Slope=(-6)/(2)`

`Slope=-3`

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