Problem Solving PDF

CHAPTER 3 Problem Solving Copyright © Cengage Learning. All rights reserved.  3.2 Section Problem Solving with Patterns...

CHAPTER 3 Problem Solving Copyright © Cengage Learning. All rights reserved.  3.2 Section Problem Solving with Patterns Copyright © Cengage Learning. All rights reserved. Terms of a Sequence 3 Terms of a Sequence An ordered list of numbers such as 5, 14, 27, 44, 65,... is called a sequence. The numbers in a sequence that are separated by commas are the terms of the sequence. In the above sequence, 5 is the first term, 14 is the second term, 27 is the third term, 44 is the fourth term, and 65 is the fifth term. The three dots “...” indicate that the sequence continues beyond 65, which is the last written term. It is customary to use the subscript notation an to designate the nth term of a sequence. 4 Terms of a Sequence We often construct a difference table, which shows the differences between successive terms of the sequence. The following table is a difference table for the sequence 2, 5, 8, 11, 14,... Each of the numbers in row (1) of the table is the difference between the two closest numbers just above it (upper right number minus upper left number). The differences in row (1) are called the first differences of the sequence. 5 Terms of a Sequence In this case, the first differences are all the same. Thus, if we use the above difference table to predict the next number in the sequence, we predict that 14 + 3 = 17 is the next term of the sequence. This prediction might be wrong; however, the pattern shown by the first differences seems to indicate that each successive term is 3 larger than the preceding term. 6 Terms of a Sequence The following table is a difference table for the sequence 5, 14, 27, 44, 65,... In this table, the first differences are not all the same. In such a situation it is often helpful to compute the successive differences of the first differences. These are shown in row (2). 7 Terms of a Sequence These differences of the first differences are called the second differences. The differences of the second differences are called the third differences. To predict the next term of a sequence, we often look for a pattern in a row of differences. 8 Example 1 – Predict the Next Term of a Sequence Use a difference table to predict the next term in the sequence. 2, 7, 24, 59, 118, 207,... Solution: Construct a difference table as shown below. 9 Example 1 – Solution cont’d The third differences, shown in blue in row (3), are all the same constant, 6. Extending row (3) so that it includes an additional 6 enables us to predict that the next second difference will be 36. Adding 36 to the first difference 89 gives us the next first difference, 125. Adding 125 to the sixth term 207 yields 332. Using the method of extending the difference table, we predict that 332 is the next term in the sequence. 10 nth-Term Formula for a Sequence 11 nth-Term Formula for a Sequence In Example 1 we used a difference table to predict the next term of a sequence. In some cases we can use patterns to predict a formula, called an nth-term formula, that generates the terms of a sequence. 12 Example 2 – Find an nth-Term Formula Assume the pattern shown by the square tiles in the following figures continues. a. What is the nth-term formula for the number of tiles in the nth figure of the sequence? b. How many tiles are in the eighth figure of the sequence? c. Which figure will consist of exactly 320 tiles? 13 Example 2 – Solution a. Examine the figures for patterns. Note that the second figure has two tiles on each of the horizontal sections and one tile between the horizontal sections. The third figure has three tiles on each horizontal section and two tiles between the horizontal sections. The fourth figure has four tiles on each horizontal section and three tiles between the horizontal sections. 14 Example 2 – Solution cont’d Thus the number of tiles in the nth figure is given by two groups of n plus a group of n less one. That is, an = 2n + (n – 1) an = 3n – 1 b. The number of tiles in the eighth figure of the sequence is 3(8) – 1 = 23. c. To determine which figure in the sequence will have 320 tiles, we solve the equation 3n – 1 = 320. 15 Example 2 – Solution cont’d 3n – 1 = 320 3n = 321 Add 1 to each side. n = 107 Divide each side by 3. The 107th figure is composed of 320 tiles. 16