Transcript Sequences - Finding a rule

A sequence is a set of terms, in a
definite order, where the terms are
obtained by some rule.
A finite sequence ends after a
certain number of terms.
An infinite sequence is one that
continues indefinitely.
For example: 1, 3, 5, 7, …
(This is a sequence of odd numbers)
1st term =
2nd term =
3rd term =
2x3–1
=1
=3
=5
nth term =
2xn–1
= 2n - 1
.
.
.
2x1–1
2x2–1
.
.
.
+2
+2
NOTATION
1st term =
2nd term =
3rd term =
.
.
.
u1
u2
u3
.
.
.
nth term =
un
OR
1st term =
2nd term =
3rd term =
.
.
.
u0
u1
u2
.
.
.
nth term =
un-1
FINDING
THE FORMULA FOR
THE TERMS OF
A SEQUENCE
A recurrence relation defines the
first term(s) in the sequence and the
relation between successive terms.
For example: 5, 8, 11, 14, …
u1
u2
u3
=5
= u1 +3
= u2 +3
.
.
.
un+1 = un +3
=8
= 11
= 3n + 2
What to look for
when looking for the rule
defining a sequence
Constant difference: coefficient of n is the difference
2nd level difference: compare with square numbers
(n
2
= 1, 4, 9, 16, …)
3rd level difference: compare with cube numbers
(n3 = 1, 8, 27, 64, …)
None of these helpful: look for powers of numbers
n-1
(2 = 1, 2, 4, 8, …)
k
k
Signs alternate: use (-1) and (-1)
-1 when k is odd
+1 when k is even
EXAMPLE:
Find the next three terms in the sequence 5, 8, 11, 14, …
EXAMPLE:
The nth term of a sequence is given by x n =
a)
1
__
2
n
Find the first four terms of the sequence.
1
____
1024
b)
Which term in the sequence is
?
c)
Express the sequence as a recurrence relation.
EXAMPLE:
Find the nth term of the sequence +1, -4, +9, -16, +25, …
EXAMPLE:
A sequence is defined by a recurrence relation of the form:
M n + 1 = aM + b.
Given that M1 = 10, M2 = 20, M 3 = 24, find the value of a and the
value of b and hence find M4 .
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