Answer:
a. ![y = 2x](https://tex.z-dn.net/?f=y%20%3D%202x)
b. The slope represents the common difference
c. ![a_n = 2n](https://tex.z-dn.net/?f=a_n%20%3D%202n)
d. ![a_{10} = 20](https://tex.z-dn.net/?f=a_%7B10%7D%20%3D%2020)
Step-by-step explanation:
Given:
The graph
<u>Solving (a): The equation</u>
First, we pick any two corresponding points on the graph
![(x_1,y_1) = (1,2)](https://tex.z-dn.net/?f=%28x_1%2Cy_1%29%20%3D%20%281%2C2%29)
![(x_2,y_2) = (2,4)](https://tex.z-dn.net/?f=%28x_2%2Cy_2%29%20%3D%20%282%2C4%29)
Next, we calculate the slope using:
![m = \frac{y_1 - y_2}{x_1 - x_2}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By_1%20-%20y_2%7D%7Bx_1%20-%20x_2%7D)
![m = \frac{2-4}{1-2}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B2-4%7D%7B1-2%7D)
![m = \frac{-2}{-1}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B-2%7D%7B-1%7D)
![m=2](https://tex.z-dn.net/?f=m%3D2)
The equation is then calculated as:
![y - y_1 = m(x - x_1)](https://tex.z-dn.net/?f=y%20-%20y_1%20%3D%20m%28x%20-%20x_1%29)
![y - 2 = 2(x - 1)](https://tex.z-dn.net/?f=y%20-%202%20%3D%202%28x%20-%201%29)
![y - 2 = 2x - 2](https://tex.z-dn.net/?f=y%20-%202%20%3D%202x%20-%202)
Make y the subject
![y = 2x - 2+2](https://tex.z-dn.net/?f=y%20%3D%202x%20-%202%2B2)
![y = 2x](https://tex.z-dn.net/?f=y%20%3D%202x)
<u>(b) Interpret the slope</u>
In (a) above, the slope is calculated as: ![m=2](https://tex.z-dn.net/?f=m%3D2)
This represents the common difference of the sequence;
<u>(c) Represent as an arithmetic sequence</u>
We have the following points from the graph:
![(x_1,y_1) = (1,2)](https://tex.z-dn.net/?f=%28x_1%2Cy_1%29%20%3D%20%281%2C2%29)
![(x_2,y_2) = (2,4)](https://tex.z-dn.net/?f=%28x_2%2Cy_2%29%20%3D%20%282%2C4%29)
![(x_3,y_3) = (3,6)](https://tex.z-dn.net/?f=%28x_3%2Cy_3%29%20%3D%20%283%2C6%29)
This means that: The first term is 2; the second is 4, the third is 6.....
So, we have:
--- First Term
--- Difference
The nth term of an AP is:
![a_n = a_1 + (n - 1)d](https://tex.z-dn.net/?f=a_n%20%3D%20a_1%20%2B%20%28n%20-%201%29d)
This gives:
![a_n = 2 + (n - 1)*2](https://tex.z-dn.net/?f=a_n%20%3D%202%20%2B%20%28n%20-%201%29%2A2)
![a_n = 2 + 2n - 2](https://tex.z-dn.net/?f=a_n%20%3D%202%20%2B%202n%20-%202)
Collect Like Terms
![a_n = - 2+2 + 2n](https://tex.z-dn.net/?f=a_n%20%3D%20-%202%2B2%20%2B%202n)
![a_n = 2n](https://tex.z-dn.net/?f=a_n%20%3D%202n)
<u>(d) Find a10</u>
To do this, we simply substitute 10 for n in ![a_n = 2n](https://tex.z-dn.net/?f=a_n%20%3D%202n)
So, we have:
![a_{10} = 2 * 10](https://tex.z-dn.net/?f=a_%7B10%7D%20%3D%202%20%2A%2010)
![a_{10} = 20](https://tex.z-dn.net/?f=a_%7B10%7D%20%3D%2020)