Sequences Photo source chromaspiral
Pre-Cal 40S Slides January 16, 2008
Jun 14, 2015
Arithmetic and geometric sequences; implicit and recursive definitions.
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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, , ,
3n+1
4, 7, 10, 13, , ,
3, 6, 12, 24, , ,
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.
Some Definitions
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: 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
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
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
d = tn - t(n - 1)
To Find The Common Difference
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.
Geometric 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.
3, 6, 12, 24, , ,
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 terma is the first termn is the "rank" of the nth term in the sequencer is the common ratio
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