Answer:
Sequence : [tex]a_n=8(\frac{1}{2})^{n-1}[/tex]
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 [tex]a_n=k[/tex] represent the point (n,k). So, we have
[tex]a_1=8,a_2=4,a_4=1,a_5=0.5[/tex]
[tex]\frac{a_2}{a_1}=\frac{4}{8}=\frac{1}{2}[/tex]
[tex]\frac{a_5}{a_4}=\frac{0.5}{1}=\frac{1}{2}[/tex]
It is clear that the above sequence is a geometric sequence because it has common ratio [tex]\frac{1}{2}[/tex].
First term : [tex]a=8[/tex]
Common ratio : [tex]r=\frac{1}{2}[/tex]
The nth term of a G.P. is
[tex]a_n=ar^{n-1}[/tex]
[tex]a_n=8(\frac{1}{2})^{n-1}[/tex]
Therefore, the required sequence is [tex]a_n=8(\frac{1}{2})^{n-1}[/tex].
We need to find the average rate of change from n = 1 to n = 3.
[tex]a_3=8(\frac{1}{2})^{3-1}=2[/tex]
[tex]Slope=\dfrac{a_3-a_1}{3-1}[/tex]
[tex]Slope=\dfrac{2-8}{2}[/tex]
[tex]Slope=\dfrac{-6}{2}[/tex]
[tex]Slope=-3[/tex]
Therefore, the average rate of change from n = 1 to n = 3 is -3.
