Introduction to Sequences.

1. Sequences
Recursive Blanket Flower by flickr user gadl

2. 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, , ,

3. 4, 7, 10, 13, , ,

4. 3, 6, 12, 24, , ,

5. Arithmetic sequences on the calculator ...

6. 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.
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.

7. To Find The Common Difference
d = tn - t(n - 1)
d is the common difference
tn is an arbitrary term in the sequence
t(n - 1) is the term immediately before tn in the sequence
To Find the nth Term In an Arithmetic Sequence
tn = a + (n - 1)d
tn is the nth term
a is the first term
n is the quot;rankquot; of the nth term in the sequence
d is the common difference
Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ...
Solution: a = 11 t51 = 11 + (51 - 1)(-6)
d = 5 - 11 t51 = 11 + (50)(-6)
= -6 t51 = 11 - 300
n = 51 t51 = -289

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

9. 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

10. 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 or more terms to verify this answer.)