a We have been given three figures we can describe as circles divided into sectors. Let's see if they follow a pattern.
Figure
Number of Sectors
1
2
2
4
3
6
For each figure the number of sectors is two more than for the previous figure. By continuing this pattern we can determine how many sectors the next three figures must be divided in.
Figure
Number of Sectors
1
2
2
4
3
6
4
8
5
10
6
12
We are now going to draw these, one at the time. Let's begin with the circle with eight sectors.
Our next diagram will show a circle divided into ten sectors.
Our final image will be of a circle which has been divided into a total of twelve sectors.
a_n= a_1+(n-1) dWe know that with each new figure 2 sectors are being added. Therefore, our common difference is 2. We also know that since the first figure is divided into 2 sectors the first first term of the sequence is 2.
a_n= a_1+(n-1) d ⇓ a_n= 2+(n-1) 2
By substituting 20 for n in the equation, we can find out how many sectors the 20th figure has.
