c In Part A, we found that each figure repeats three times, starting with triangles. To find the number of sides of the 23rd figure, we can divide 23 by 3 to see how many times we will cycle through the pattern.
323
▼
Write fraction as a mixed number
732
As we can see,
seven cycles of
three figures will be completed. Then, the
23rd figure will be the
second figure of the eighth cycle. If we take each cycle as a , we can write an . Let's recall the for an arithmetic sequence.
A(n)=A(1)+(n−1)d
Here,
A(1) is the first term and
d is the . In this case, each cycle is one term. In the first cycle of figures, we have triangles —
3 sides. Then, we have squares —
4 sides. This pattern continues, adding
1 to the number of sides each time. Therefore, the common difference is
1. With this information, we can write the formula.
A(n)=3+(n−1)1
Let's evaluate this formula for the eighth cycle of figures, where the
23rd figure belongs.
A(n)=3+(n−1)1
A(8)=3+(8−1)1
A(8)=10
Therefore, the
23rd figure will have
10 sides.