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

Answer 2

It should be noted that the sequence that's graphed will be A. 8(1/2)ⁿ − 2; average rate of change is −6.

How to identify the sequence

The sequence of the progression have been given. The second, third, fourth, fifth terms are 8, 4, 2, and 1.

The power is decreasing by 2. Therefore, the average rate of change will be:

= (4 - 16)/2

= -12/2

= -6

In conclusion, the correct option is A.

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