1. 1.2 Mathematical Patterns Objectives: 1. Define key terms: sequence, sequence notation, recursive functions 2. Create a graph of a sequence 3. Apply sequences to real-world situations

2. Key terms: A sequence is an ordered list of numbers. Each number in the list is called a term of the sequence. An infinite sequence is a sequence with an infinite number of terms. The three dots, or points of ellipsis, at the end of the sequence indicate that the same pattern continues for an infinite number of terms. Identify the next 3 terms of each of the sequences to the left.

3. Sequence Notation The following notation denotes specific terms of a sequence: ◦ The first term of a sequence is denoted u 1. ◦ The second term u 2. ◦ The term in the n th position, called the n th term, is denoted by u n. ◦ The term before u n is u n −1. ◦ The term before u n −1 is u n −2, etc. For example, the 9 th term would be u 9 and u n −1 would be the 8 th term, which is u 9 −1 or u 8.

4. Example #1 Terms of a Sequence Continue drawing the pattern shown below by drawing the next two diagrams, and write a sequence that represents the number of circles in each diagram.

5. Example #2 Graph of a Sequence Graph the first five terms of the sequence {2, 5, 8, 11, 14, …} Make the terms into ordered pairs: (1, 2), (2, 5), (3, 8), (4, 11), (5, 14) This sequence is called an arithmetic sequence and forms a straight line when graphed.

6. Example #3 Recursively Defined Sequence Define the sequence recursively and graph it. A sequence is defined recursively if the first term is given and there is a method of determining the nth term by using the terms that precede it. This is called a geometric sequence and its graph is exponential.

7. Example #4 Using Alternate Sequence Notation A basketball is dropped from a height of 8 feet. It hits the floor and bounces to a height of 6 feet. It continues to bounce, and on each rebound it rises ¾ the height of the previous bounce. A. What would be the next term of the sequence? B. Write a recursive formula for the sequence that represents the height of the ball on each bounce. Since the initial height of 8 feet isn’t really counted as a “bounce” we set that equal to the term u 0 rather than u 1, which is instead the height of the first bounce of 6 feet.

8. Example #4 Using Alternate Sequence Notation A basketball is dropped from a height of 8 feet. It hits the floor and bounces to a height of 6 feet. It continues to bounce, and on each rebound it rises ¾ the height of the previous bounce. C. Create a table and graph showing the height of the ball on each bounce. Put calculator in sequence mode. MODE  Seq Type the recursive function into the Y = screen. The nMin is where the function begins (0 for u 0 ). Set u(nMin) = 8 for the initial height.

9. Example #4 Using Alternate Sequence Notation A basketball is dropped from a height of 8 feet. It hits the floor and bounces to a height of 6 feet. It continues to bounce, and on each rebound it rises ¾ the height of the previous bounce. C. Create a table and graph showing the height of the ball on each bounce. **Note**: Press 2 nd  7 to enter the “u” symbol and use the variable key (X, T, θ, n) for the “n”. GRAPH key brings up the graph.

10. Example #4 Using Alternate Sequence Notation A basketball is dropped from a height of 8 feet. It hits the floor and bounces to a height of 6 feet. It continues to bounce, and on each rebound it rises ¾ the height of the previous bounce. D. To the nearest tenth of a foot, what would be the height on the 5 th bounce? 2 nd  GRAPH brings up the TABLE screen. From this screen we can count the bounces… on the 5 th bounce the ball bounces 1.9 feet in the air.

11. Example #5 Salary Raise Sequence Rick owns an automobile dealership. Last year, he spent $16,000 on advertising. He plans to increase his advertising expenditures by $1200 this year and in each subsequent year. What will be the amount he spends on advertising in the sixth year? Find a recursive function to represent this problem and use a table and a graph to find the solution. Since $16,000 was spent on advertising last year, we once again set it equal to u 0 rather than u 1. Therefore, the current year amount is set to u 1.

12. Example #5 Salary Raise Sequence Rick owns an automobile dealership. This year, he has budgeted $16,000 on advertising. He plans to increase his advertising expenditures by $1200 next year and in each subsequent year. What will be the amount he spends on advertising in the sixth year? Find a recursive function to represent this problem and use a table or graph to find the solution. Since $16,000 was spent on advertising this year, we must set it equal to u 1. YearNowNext 1$16,000$16,000 + $1200 = $17,200 2$17,200$17,200 + $1200 = $18,400 3$18,400$18,400 + $1200 = $19,600 4$19,600$19,600 + $1200 = $20,800 5$20,800$20,800 + $1200 = $22,000 6$22,000$22,000 + $1200 = $23,200

13. Example #6 Sequence Formed by Adding a Pattern of Values Triangular numbers are numbers that can be represented geometrically as triangles of points, following the pattern shown below (each row has one more point than the row above it). Here are the first four triangular numbers. A. Find a recursive function to represent the n th triangular number. 13610

14. Example #6 Sequence Formed by Adding a Pattern of Values Triangular numbers are numbers that can be represented geometrically as triangles of points, following the pattern shown below (each row has one more point than the row above it). Here are the first four triangular numbers. B. Use a table to find the 20 th triangular number. 13610 210

15. Example #7 Application Ms. Long creates an endowment fund for her alma mater. She places $250,000 in the fund to start, and will add $50,000 to the fund each year. She states that 25% of the total amount in the fund shall be used each year for scholarships. How much will be in the fund at the end of the sixth year? Since $250,000 was the starting value, we set it equal to u 0 rather than u 1 as after the first year is completed $50,000 will be added and then 25% is removed for scholarships. 100% − 25% = 75%

16. Example #7 Application Ms. Long creates an endowment fund for her alma mater. She places $250,000 in the fund to start, and will add $50,000 to the fund each year. She states that 25% of the total amount in the fund shall be used each year for scholarships. How much will be in the fund at the end of the sixth year? After 6 years there will be $167,798 in the endowment fund.