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# Applied 40S May 27, 2009

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Introduction to sequences. Arithmetic sequences defined recursively and implicitly.

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### Applied 40S May 27, 2009

1. 1. Sequences Recursive Blanket Flower by ﬂickr user gadl
2. 2. At a sea port, the depth of the water, h meter, at time, t hours, during a certain day is given by this formula: REVIEW (a) State the: (i) period (ii) amplitude (iii) phase shift. (b) What is the maximum depth of the water? When does it occur?
3. 3. At a sea port, the depth of the water, h meter, at time, t hours, during a certain day is given by this formula: REVIEW (c) Determine the depth (d) Determine one time when of the water at 5:00 am the water is 2.25 meters deep. and at 12:00 noon.
4. 4. Find the next three terms in each sequence of numbers ... 4, 7, 10, 13, , , 3, 6, 12, 24, , , 32, 16, 8, 4, , , 1, 1, 2, 3, 5, 8,13, , ,
5. 5. 4, 7, 10, 13, , , Arithmetic sequences on the calculator ...
6. 6. 4, 7, 10, 13, , , Sum of Terms & Graphing
7. 7. 4, 7, 10, 13, , , Graphing Sequences & Implicit Equations
8. 8. Sequence: An ordered list of numbers that follow a certain pattern (or rule). Arithmetic Sequence: (i) Recursive Deﬁnition: An ordered list of numbers generated by continuously adding a value (the common difference) to a given ﬁrst term. (ii) Implicit Deﬁnition: An ordered list of numbers where each number in the list is generated by a linear equation. Example:
9. 9. Sequence: An ordered list of numbers that follow a certain pattern (or rule). Common Difference (d): (i) The number that is repeatedly added to successive terms in an arithmetic sequence. (ii) From the implicit deﬁnition, d is the slope of the linear equation. Example: 4, 7, 10, 13, , ,
10. 10. Determine which of the following sequences are arithmetic. If a sequence is arithmetic, write the values of a and d. HOMEWORK (a) 5, 9, 13, 17, ... (b) 1, 6, 10, 15, 19, ... Given the values of a and d, write the ﬁrst 5 terms of each arithmetic sequence. (a) a = 7, d, = 2 (b) a = -4, d, = 6
11. 11. List the ﬁrst 4 terms of the sequence determined by each of the following implicit deﬁnitions. HOMEWORK
12. 12. Use your calculator to ﬁnd the ﬁrst 10 terms and the sum of the ﬁrst 10 terms of the sequence: 16, 8, 4, 2, . . . HOMEWORK (a) What is the 10th term? What is the sum of the ﬁrst 10 terms? (b) Extend the sequence to 15 terms. What is the 15th term? What is the sum of 15 terms? (c) What happens to the terms as you have more terms? Also, what happens to the value of the sum of the terms as you have more terms? (Look at 30, 50, or more terms to verify this answer.)