Date post: | 20-Jun-2015 |
Category: |
Education |
Upload: | darren-kuropatwa |
View: | 1,677 times |
Download: | 0 times |
Sequences all around us
patterns warped and otherwise by flickr user Grant MacDonald
Find the next three terms in each sequence of numbers ...
1, 1, 2, 3, 5, 8,13, , ,
3, 6, 12, 24, , ,
4, 7, 10, 13, , ,
32, 16, 8, 4, , ,
4, 7, 10, 13, , , 16 19 22
RANK
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, , ,
To Find The Common Difference
d is the common differencetn is an arbitrary term in the sequencet(n - 1) is the term immediately before tn in the sequence
d = tn - t(n - 1)
Example: Find the common difference for the sequence:
11, 5, -1, -7, ...
5 - 11= -6
-1 - 5 = -6
-7 - (-1) = -6
d = -6
Solution: a = 11 t51 = 11 + (51 - 1)(-6) d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289
Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ...
tn is the nth terma is the first termn is the "rank" of the nth term in the sequenced is the common difference
tn = a + (n - 1)d
To Find the nth Term In an Arithmetic Sequence
3, 6, 12, 24, , ,
3, 6, 12, 24, , ,
Geometic Sequence:
(ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by an exponential equation.
(i) Recursive Definition: An ordered list of numbers generated by continuously multiplying a value (the common ratio) with a given first term.
Common Ratio (r):
(ii) From the implicit definition, r is the base of the exponential function.
(i) The number that is repeatedly multiplied to successive terms in a geometic sequence.
To Find The Common Ratio
t(n - 1) is the term immediately before tn in the sequence
tn is an arbitrary term in the sequence
r is the common ratio
To Find the nth Term In a Geometic Sequence
r is the common ratio
n is the "rank" of the nth term in the sequence
a is the first term
tn is the nth term
32, 16, 8, 4, , ,
Write the implicit definition for this sequence.
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: