Question

Identify the sequence graphed below and the average rate of change from n = 0 to n = 2.

coordinate plane showing the point 1, 10, point 2, 5, and point 4, 1.25

Answer 1

We must find the equation of this function first.  This is a geometric sequence as each term is a constant ratio of the previous term...

5/10=1/2, And this sequence can be expressed as:

a(n)=10(1/2)^(n-1)  So the term when n=0 would be 20

The average rate of change is just the change in y divided by the change in x, in this case:

(20-5)/(2-0)

7.5

Answer 2

Answer:

The sequence is:

         $a_n=10\times (\dfrac{1}{2})^{n-1}$

and the average rate of change from n=0 to n=2 is:

                           -7.5

Step-by-step explanation:

We are given the points as:

(1,10), (2,5) and (4,1.25)

Clearly after looking the point we see that these point follow a geometric sequence.

Since with the increase in x-value by 1 unit there is a decrease in the y-value by a factor of 1/2

Let the points be denoted by: $(n,a_n)$

Hence, we have the sequence as:

$a_n=10\times (\dfrac{1}{2})^{n-1}$

Since, when x=n=1 we have:

$y=a_n=10\times (\dfrac{1}{2})^{1-1}\\\\\\y=a_n=10\times (\dfrac{1}{2})^{0}\\\\\\y=a_n=10\times 1\\\\\\y=a_n=10$

Similarly we can check the other points as well

When x=n=0

we have:

$y=a_n=10\times (\dfrac{1}{2})^{0-1}=10\times (\dfrac{1}{2})^-1\\\\y=a_n=10\times 2\\\\y=a_n=20$

Now, the average rate of change from n=0 to n=2 is calculated as:

$Rate=\dfrac{a_2-a_0}{2-0}\\\\\\Rate=\dfrac{a_2-a_0}{2}\\\\\\Rate=\dfrac{5-20}{2}\\\\\\Rate=\dfrac{-15}{2}\\\\\\Rate=-7.5$

Hence, Rate= -7.5