1 / 16

1.2 Mathematical Patterns

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

tolla
Download Presentation

1.2 Mathematical Patterns

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 1.2 Mathematical Patterns Objectives: Define key terms: sequence, sequence notation, recursive functions Create a graph of a sequence 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 u1. • The second term u2. • The term in the nth position, called the nth term, is denoted by un. • The term before un is un−1. • The term before un−1 is un−2 , etc. For example, the 9th term would be u9 and un−1 would be the 8th term, which is u9−1 or u8.

  4. Example #1Terms 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 #2Graph 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 #3Recursively 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 #4Using 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. • What would be the next term of the sequence? • 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 u0 rather than u1, which is instead the height of the first bounce of 6 feet.

  8. Example #4Using 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. • Create a table and graph showing the height of the ball on each bounce. Type the recursive function into the Y = screen. The nMin is where the function begins (0 for u0). Set u(nMin) = 8 for the initial height. Put calculator in sequence mode. MODESeq

  9. Example #4Using 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. • Create a table and graph showing the height of the ball on each bounce. GRAPH key brings up the graph. **Note**: Press 2nd 7 to enter the “u” symbol and use the variable key (X, T, θ, n) for the “n”.

  10. Example #4Using 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. • To the nearest tenth of a foot, what would be the height on the 5th bounce? 2nd GRAPH brings up the TABLE screen. From this screen we can count the bounces… on the 5th bounce the ball bounces 1.9 feet in the air.

  11. Example #5Salary 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 u0 rather than u1. Therefore, the current year amount is set to u1.

  12. Example #5Salary 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 u1.

  13. Example #6Sequence 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. • Find a recursive function to represent the nth triangular number. 1 3 6 10

  14. Example #6Sequence 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. • Use a table to find the 20th triangular number. 1 3 6 10 210

  15. Example #7Application • 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 u0 rather than u1 as after the first year is completed $50,000 will be added and then 25% is removed for scholarships. 100% − 25% = 75%

  16. Example #7Application • 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.

More Related