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

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, , ,

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

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:

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, , ,

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

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 \"rank\" 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

3, 6, 12, 24, , ,

3, 6, 12, 24, , ,

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.

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.

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

To Find the nth Term In a Geometic Sequence
tn is the nth term
a is the first term
n is the \"rank\" of the nth term in the sequence
r is the common ratio

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

Some \"quickies\" 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: