Sequences

Recursive Blanket Flower by flickr user gadl

At a sea port, the depth of the water, h meter, at time, t hours, during a certain day is given by this formula:

(a) State the: (i) period (ii) amplitude (iii) phase shift.

(b) What is the maximum depth of the water? When does it occur?

REVIEW

At a sea port, the depth of the water, h meter, at time, t hours, during a certain day is given by this formula:

(c) Determine the depth of the water at 5:00 am and at 12:00 noon.

(d) Determine one time when the water is 2.25 meters deep.

REVIEW

Find the next three terms in each sequence of numbers ...

1, 1, 2, 3, 5, 8,13, , ,

32, 16, 8, 4, , ,

3, 6, 12, 24, , ,

4, 7, 10, 13, , ,

4, 7, 10, 13, , ,

Arithmetic sequences on the calculator ...

4, 7, 10, 13, , ,

Sum of Terms & Graphing

4, 7, 10, 13, , ,

Graphing Sequences & Implicit Equations

Sequence: An ordered list of numbers that follow a certain pattern (or rule).

Arithmetic Sequence:

Example:

(ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by a linear equation.

(i) Recursive Definition: An ordered list of numbers generated by continuously adding a value (the common difference) to a given first term.

Sequence: An ordered list of numbers that follow a certain pattern (or rule).

(ii) From the implicit definition, d is the slope of the linear equation.

(i) The number that is repeatedly added to successive terms in an arithmetic sequence.

Common Difference (d):

Example: 4, 7, 10, 13, , ,

Determine which of the following sequences are arithmetic. If a sequence is arithmetic, write the values of a and d.

(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

HOMEWORK

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

Use your calculator to find the first 10 terms and the sum of the first 10 terms of the sequence: 16, 8, 4, 2, . . .

(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.)

HOMEWORK