pre-cal 40s slides may 29, 2007
DESCRIPTION
Introductions to geometric sequences beginning with arithmetic sequences.TRANSCRIPT
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, , ,
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.
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 terma is the first termn is the "rank" of the nth term in the sequenced 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
32, 16, 8, 4, , ,
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 terma is the first termn is the "rank" of the nth term in the sequencer is the common ratio