Pearson Algebra 1 Common Core, 2011
PA
Pearson Algebra 1 Common Core, 2011 View details
7. Arithmetic Sequences
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Exercise 75 Page 281

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 by to see how many times we will cycle through the pattern.
Write fraction as a mixed number
As we can see, cycles of the color pattern will be completed. Moreover, the twentieth figure will have the color of the 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 figure, we can divide by to see how many times we will cycle through the pattern.
Write fraction as a mixed number
As we can see, cycles of figures will be completed. Then, the figure will be the 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.
Here, is the first term and is the common difference. In this case, each cycle is one term. In the first cycle of figures, we have triangles — sides. Then, we have squares — sides. This pattern continues, adding to the number of sides each time. Therefore, the common difference is With this information, we can write the formula.
Let's evaluate this formula for the eighth cycle of figures, where the figure belongs.
Evaluate right-hand side
Therefore, the figure will have sides.