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Mathematical Patterns
1
2
3
Definitions
Explicit & Recursive Formulas
Practice Problems
Term Definitions
2






a1 is the first term
a2 is the second term
a3 is the third term
an is a given term (generic)
an-1 is the previous term
an+1 is the next term
Definitions
3




Sequence
 A list of numbers in a particular order
Term
 Each number in the sequence
Recursive Formula
 Relates each term after the first term to the pme
before it
Explicit Formula
 Describes the nth term of a sequence using the
number n
Explicit Formulas
4


Substitute the number in the “n” position (subscript)
of an into the formula for each term.
Example: In the sequence 2,4,6,8,10…, the nth term
is twice the value of n
2
a1
So…
4
6
a3
a2
an  2n
8
a4
10
a5
Explicit Formulas Cont.
5
n
nth term
1
2
3
4
5
a1 = 2(1) = 2
a2 = 2(2) = 4
a3 = 2(3) = 6
a4 = 2(4) = 8
a5 = 2(5) = 10
Generating a Sequence Using an
Explicit Formula
6
A sequence has an explicit formula an = 3n – 2.
 What are the first 5 terms of the sequence
n
an

1
2
3
4
5
a1
a2
 3(1)  2
 3(2)  2
1
4
a3
 3(3)  2
7
a4
a5
 3(4)  2
 3(5)  2
 10
 13
Writing an Explicit Formula
7



Look for a pattern between “n” and “an”
Use “n” to express the relationship between “n” and
“an”
Can be anything, however the following forms are
common:
an  x ( n)  y
an  x ( n)  y
z
Writing an Explicit Formula Example
8
Write an explicit formula for the sequence:
7
10
13
16
n2
n3
n4
n5
4
n 1
an  3n
gets us close
an  3n  1
Writing a Recursive Formula
9


Subtract consecutive terms to find out what happens
from one term to the next
Use “n” to express the relationship between
successive terms
an  an1  ?

Solve for “n”
Writing a Recursive Formula Example
10
Write a recursive formula for the sequence 1,3,6,10,15,21
a2  a1  3  1  2
a3  a2  6  3  3
a4  a3  10  6  4
a5  a4  15  10  5
an  an 1  ?
an  an 1  n
 an 1  an 1
an  an 1  n