a Pay attention to how the number of yellow folders is changing from one step to the next one.
B
b The number of green folders can be obtained as the total number of folders minus the number of yellow ones.
C
c Use a rational function written in terms of your previous results.
A
aExample model: Y(n) = 1 +4(n-1)
B
bExample model: G(n) = 4(n-1)^2
C
cExample model: R(n) = 1+4(n-1)/4(n-1)^2
Practice makes perfect
a To find the pattern we should pay attention to how the number of yellow folders is changing in each step. Notice that the yellow folders are increasing by one on each diagonal. In consequence, the number of yellow folders is increasing by 4 in each step, starting from 1.
Since the yellow folders are increasing at a constant rate of 4, we can model this by using a linear function with a slope of 4. Furthermore, since it starts with 1 folder, the y-intercept should be 1.
Y(n) = 1 + 4(n)
However, we want n to start at 1 and not at 0. We can fix this it replacing n-1 for n in the function rule.
Y(n) = 1 + 4(n-1)
To verify if our model works, we can find the predicted number of yellow folders for n= 1, 2, and 3 and count them in the figure to compare.
n
1+4(n-1)
Y(n)
1
1+4(n- 1)
Y( 1)=1
2
1+4(n- 2)
Y( 2)=5
3
1+4(n- 3)
Y( 3)=9
We can see that the predictions match the number of yellow folders at each step, so we can be sure that our model is correct.
Please note that there are different ways of writing a function equivalent to this one.
b Note that the total of folders at each step form a squared arrangement, increasing in side by 2 folders each time. Since it starts from 1, the number of folders per side in the arrangement is always an odd number. We can represent an arbitrary odd number by using the expression 2n-1, where n can be any natural number.
1 → 3 → 5 → ... → 2n-1
Furthermore, since the arrangement is square, the total number of folders T(n) can be obtained by multiplying the number of folders per side times itself.
T(n) = (2n-1)(2n-1)=(2n-1)^2
Also, we know from Part A that the number of yellow folders is Y(n)=1+4(n-1). Hence, the difference between both functions represents the number of green folders.
To verify if our model works, we can find the predicted number of green folders for n= 1, 2, and 3 and count them in the figure to compare.
n
4(n-1)^2
G(n)
1
4( 1-1)^2
G( 1)=0
2
4( 2-1)^2
G( 2)=4
3
4( 3-1)^2
G( 3)=16
We can see that the predictions match the number of green folders at each step, so we can be sure that our model is correct.
Please note that there are different ways of writing a function equivalent to this one.
c To find a model for the ratio of yellow folders to green folders R(n), we can use a rational function whose numerator is the polynomial function Y(n) found in Part A and its denominator the polynomial function G(n) found in Part B.
R(n) = Y(n)/G(n) = 1 + 4(n-1)/4(n-1)^2Since the next step is Step 4, we can substitute n=4 to obtain the ratio of yellow folders to green folders in that step.
Recall that in Part A we saw that the pattern was growing by one yellow folder on each diagonal, and the rest of spaces are taken by green folders. With this in mind, we can draw the diagram for Step 4. Then, we can count the folders and verify our prediction.
As we can see, the number of yellow folders is 13 and the number of green folders is 36. The ratio of yellow folders to green folders is then 1336= 13, as predicted.
