Warm-up
Finding Terms of a Sequence
1. Find the next four terms in the sequence.
1, 1, 2, 3, 5, 8, __, __, ___, ____,…
2. Write the explicit formula for the sequence.
5, 13, 21, 29, 37, ___, ___, ___, ___,…
3. Write the explicit formula for the sequence.
800, 400, 200, 100, __, __, __, ___,…
4. Write the explicit formula for the sequence.
-27, 9, -3, 1, -1/3, __, __, __, __,...
Answers
Lesson 8.3
Recursive Sequences
Objectives: 1. Find particular terms of sequence
from the given general term.
2. Use recursion formulas to find
subsequent terms.
3. Determine a formula from a
sequence of numbers.
What is a
recursive sequence?
Definition:
A recursive sequence is the process in which
each step of a pattern is dependent on the
step or steps before it.
Recursion Formulas
A recursion formula defines the nth term
of a sequence as a function of the
previous term. If the first term of a
sequence is known, then the recursion
formula can be used to determine the
remaining terms.
Sequence and Terms
Let’s look at the following sequence
Do you know what the rule is for the sequence?
n²
1, 4, 9, 16, 25, 36, 49, …,
a1
a2
a3
a4
a5
a6
a7
The letter a with a subscript is used to represent
function values of a sequence.
The subscripts identify the location of a term.
How to read the subscripts:
an 1
the prior
term
an
a term in
the
sequence
an 1
the next
term
Ex. 1: Find the first four terms of the
sequence:
General Term
a1  5
an  3an1  2
Let’s be sure we understand what is given
a1  5
an

3an 1 + 2
is
The first
term is 5
Each term
after the
first
3 times the
previous
term
Plus 2
Continued…
Ex. 1: Find the first four terms of the sequence:
a1  5
an  3an1  2
Start with general term for n>1
n=1
a1  5
n=2
a2  3a21  2  3a1  2  3(5)  2  15  2  17
n=3
a3  3a31  2  3a2  2  3(17)  2  51  2  53
n=4
a4  3a41  2  3a3  2  3(53)  2  159  2  161
given
Answer = 5, 17, 53, 161
Your turn:
Ex 2: Find the next four terms of the sequence.
an  2an1
a1  3
Start with general term for n>1
n=1
a1  3
given
n=2
a2  2a21
 2a1
 2(3)  6
n=3
a3  2a31
 2a2
 2(6)  12
n=4
a4  2a41
 2a3
 2(12)  24
Answer = 3, 6, 12, 24
Try another…
an  4an1  2an2
a1  2 a2  1
n=1
a1  2
given
given
a

1
n=2
2
n=3 a  4a
3
31  2a3 2 = 4a2  2a1
n=4 a  4a
4
41  2a42 = 4a3  2a2
n=5 a  4a
5
51  2a5 2 = 4a4  2a3
4–4=0
0 – 2 = -2
-8 – 0 = -8
Answer = 2, 1, 0, -2, -8
Your turn
Write a recursive formula for the sequences below.
Step 1 : Determine if it is arithmetic or geometric.
Step 2 : Plug in to either the geometric or arithmetic recursive formula.
Step 3 : Make sure you tell us what a1 is equal to.
Arithmetic
an  an 1  d
Ex. 3
Geometric
an  r  an 1
a1  _____
a1  __
Ex. 4
3, 6, 12, 24, 48, …
7, 3, -1, -5, -9, …
The common difference = -4
an  an1  4
a1  7
The common ratio = 2
The first
term = 7
an  2an1
a1  3
The first
term = 3
Last Example…
Choose the recursive formula for the given
sequence.
Answer = C
Summary:
What is a recursive sequence?
Homework:
Worksheet 8.3 and quest review