Question

Identify the sequence graphed below and the average rate of change from n = 1 to n = 3.

Answer 1

Answer:

The required sequence is $a_n=20(\frac{1}{2})^{n-1}$. The average rate of change from n = 1 to n = 3 is -7.5.

Step-by-step explanation:

From the given graph it is clear that the sequence is a GP because the all terms are half of their previous terms.

Here, $a_2=10,a_3=5,a_4=2.5,a_5=1.25$

$r=\frac{a_3}{a_2}=\frac{5}{10}=\frac{1}{2}$

The common ratio of GP is 1/2.

$r=\frac{a_2}{a_1}$

$\frac{1}{2}=\frac{10}{a_1}$

$a_1=20$

The first term of the sequence is 20.

The formula for sequence is

$a_n=a(r)^{n-1}$

Where a is first term and r is common difference.

The required sequence is

$a_n=20(\frac{1}{2})^{n-1}$

The formula for rate of change is

$m=\frac{f(x_2)-f(x_1)}{x_2-x_1}$

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

$m=\frac{f(3)-f(1)}{3-1}$

$m=\frac{5-20}{3-1}$

$m=\frac{-15}{2}=-7.5$

Therefore the required sequence is $a_n=20(\frac{1}{2})^{n-1}$. The average rate of change from n = 1 to n = 3 is -7.5.

Answer 2

Answer:

y = 20*(1/2)^(n-1)

average rate of change = -7.5

Step-by-step explanation:

In the graph, the point (1, 20) is missing.

This kind of plot in which every value of the function is a half of the previous one (10, 5, 2.5, 1.25) has the next form: y = a*(1/2)^(n-1). From here is deduced that for n = 1, y = 20, so that, for the next n-value (2), y = 20/2 = 10.  

Replacing n = 1  in the formula gives:

y = a*(1/2)^(1-1)

y = a = 20

So,  y = 20*(1/2)^(n-1)

The average rate of change from n = 1 to n = 3 is calculated taking into account y-values at those n-values, as follows:

average rate of change = (y3 - y1)/(n3 - n1) =  (5 - 20)/(3 - 1) = -7.5

It coincides with the slope of the line which intercepts those points.