Introduction to arithmetic and geometric sequences.

1. Sequences
all around us
patterns warped and otherwise
by flickr user Grant MacDonald

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. RANK
4, 7, 10, 13, 16 , 19 , 22

4. 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:

5. 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, , ,

6. To Find The Common Difference
d is the common difference
tn is an arbitrary term in the sequence
d = tn - t(n - 1)
t(n - 1) is the term immediately before tn
in the sequence
Example: Find the common difference for the sequence:
11, 5, -1, -7, ...
5 - 11= -6
-1 - 5 = -6 d = -6
-7 - (-1) = -6

7. To Find the nth Term In an
Arithmetic Sequence
t is the nth term
n
t = a + (n - 1)d a is the first term
n 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. 3, 6, 12, 24, , ,

9. 3, 6, 12, 24, , ,

10. Geometic Sequence:
(i) Recursive Definition: An ordered list of numbers generated by
continuously multiplying a value (the common ratio) with a given
first term.
(ii) Implicit Definition: An ordered list of numbers where each
number in the list is generated by an exponential equation.

11. Common Ratio (r):
(i) The number that is repeatedly multiplied to successive terms in
a geometic sequence.
(ii) From the implicit definition, r is the base of the exponential
function.

12. To Find The Common Ratio
r is the common ratio
tn is an arbitrary term in the sequence
t(n - 1) is the term immediately before tn in the sequence

13. To Find the nth Term In a Geometic Sequence
tn is the nth term
a is the first term
n is the quot;rankquot; of the nth term in the sequence
r is the common ratio

14. Write the implicit definition for this sequence.
32, 16, 8, 4, , ,

15. Some quot;quickiesquot; to get us started ...
Find the value(s) of r in .
In the geometric sequence, if = 3 and r = 2 , find .
If the first term of a geometric progression is and the common ratio
is -3, find the next three terms.
Determine the common ratio for the geometric sequence: