12.5 Recursive Rules with
Sequences & Functions
Alg II
Explicit Rule
• A function based on a term’s position, n, in
a sequence.
• All the rules for the nth term that we’ve
been working with are explicit rules; such
as an=a1rn-1.
Recursive Rule
• Gives the first term(s) of a sequence and
an equation that relates the given term(s)
to the next terms in the sequence.
• For example: Given a0=1 and an=an-1-2
• The 1st five terms of this sequence would
be: a0, a1, a2, a3, a4 OR
• 1, -1, -3, -5, -7
Ex. 1: Write the 1st 5 terms of the
sequence.
• a1=2, a2=2, an=an-2-an-1
1st
nd term
2
term
a3=a3-2-a3-1=a1-a2=2-2=0
a4=a4-2-a4-1=a2-a3=2-0=2
a5=a5-2-a5-1=a3-a4=0-2=-2
2, 2, 0, 2, -2
Recursive Equations for…
Ex. 2: Write the indicated rule for the
arithmetic sequence with a1=15 and d=5.
• Explicit rule
an=a1+(n-1)d
an=15+(n-1)5
an=15+5n-5
an=10+5n
• Recursive rule
(*Use the idea that you get
the next term by adding
5 to the previous term.)
Or an=an-1+5
So, a recursive rule would
be a1=15, an=an-1+5
Ex. 3: Write the indicated rule for the
geometric sequence with a1=4 and r=0.2.
• Explicit rule
an=a1rn-1
an=4(0.2)n-1
• Recursive rule
(*Use the idea that you get the next
term by multiplying the previous
term by 0.2)
Or an=r*an-1=0.2an-1
So, a recursive rule for the sequence
would be a1=4, an=0.2an-1
12-5B Recursive Rules with
Sequences & functions
•
•
Iteration
Ex. 2 Write a recursive rule for the
sequence 1,2,2,4,8,32,… .
• First, notice the sequence is neither arithmetic
nor geometric.
• So, try to find the pattern.
• Notice each term is the product of the previous
2 terms.
• Or, an-1*an-2
• So, a recursive rule would be:
a1=1, a2=2, an= an-1*an-2
Ex. 3: Write a recursive rule for the
sequence 1,1,4,10,28,76.
• Is the sequence arithmetic, geometric, or
neither?
• Find the pattern.
• 2 times the sum of the previous 2 terms
• Or 2(an-1+an-2)
• So the recursive rule would be:
a1=1, a2=1, an= 2(an-1+an-2)
Assignment