Introduction to sequences. Arithmetic sequences defined recursively and implicitly.

1. Sequences
Recursive Blanket Flower by flickr user gadl

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. 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. 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. 4, 7, 10, 13, , ,
Arithmetic sequences
on the calculator ...

6. 4, 7, 10, 13, , ,
Sum of Terms & Graphing

7. 4, 7, 10, 13, , ,
Graphing Sequences & Implicit Equations

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. 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. 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. List the first 4 terms of the sequence determined by each of the
following implicit definitions. HOMEWORK

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.)