Question

Identify the sequence graphed below and the average rate of change from n = 1 to n = 3. coordinate plane showing the point 2, 8, point 3, 4, point 4, 2, and point 5, 1.
Answer choices:
an = 8(1/2)^n − 2; average rate of change is −6
an = 10(1/2)^n − 2; average rate of change is 6
an = 8(1/2)^n − 2; average rate of change is 6
an = 10(1/2)^n − 2; average rate of change is −6

Answer 1

Let $a_n$ be the n'the term of the sequence.


The first term of the sequence is $a_1$

the second $a_2$ is 8

the third term  $a_3$ is 4

the fourth term $a_4$ is 2

the fifth  term $a_5$ is 1


We notice that:

i) the values of the terms are powers of 2, and decreasing:

$a_2=8=2^3\\\\a_3=4=2^2\\\\a_4=2=2^1\\\\a_5=1=2^0\\\\.\\\\.$

ii) we also notice that the subscript integer of a, and the power of 2 at that term, add  to 5.

thus the general term of the sequence is given by:
              
                                                  $a_n=2^{(5-n)}$.

to fit the choices of the question, we can modify this formula as follows:

$a_n=2^{(5-n)}=2^{3-(n-2)}=2^3\cdot2^{-(n-2)}=8\cdot (\frac{1}{2})^{(n-2)}$


The average rate of change from n=1 to n=3 is defined as:

$\frac{a_3-a_1}{3-1}= \frac{a_3-a_1}{2}$


$a_1$ can be found using the general formula, 

$a_1=2^{(5-1)}=2^{4}=16$,

thus the average rate of change is  (4-16)/2=-12/2=-6



Answer:

 $a_n=8\cdot (\frac{1}{2})^{(n-2)}$,

 average rate of change=-6