a Pay attention to the pattern to see how it is changing.
B
b Recall the pattern characteristics from Part A. How many times does each figure repeat? How many colors appear in the color pattern? In which order do they appear?
C
c How many times does each figure repeat? How is the number of sides changing? Can you do something so that the pattern becomes an arithmetic sequence?
A
a
B
bColor of the Figure: Blue
Explanation: See solution.
C
cNumber of Sides: 10
Explanation: See solution.
Practice makes perfect
a We are given a pattern and want to determine what would be the next figure. Let's start by paying close attention to the pattern to see how it is changing.
Let's point out the characteristics of the pattern.
Each figure repeats three times.
The color pattern is red, blue, and purple, in that order.
After three similar figures appear, the next figure will have one more side than the previous one. First we have triangles, then squares, and finally pentagons.
With these features in mind, we can conclude that the next figure should be a blue pentagon.
b In Part A, we found that each figure repeats three times following the color pattern red, blue, and purple. To find the color of the twentieth term, we can divide 20 by 3 to see how many times we will cycle through the pattern.
As we can see, six cycles of the three-color pattern will be completed. Moreover, the twentieth figure will have the second color of the three-color pattern. Therefore, the twentieth figure will be blue.
c In Part A, we found that each figure repeats three times, starting with triangles. To find the number of sides of the 23^\text{rd} figure, we can divide 23 by 3 to see how many times we will cycle through the pattern.
As we can see, seven cycles of three figures will be completed. Then, the 23^(rd) figure will be the second figure of the eighth cycle. If we take each cycle as a term, we can write an arithmetic sequence. Let's recall the explicit formula for an arithmetic sequence.
A(n) = A(1) + (n-1) d
Here, A(1) is the first term and d is the common difference. 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 23^(rd) figure belongs.
