Warm-upFinding 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 8.3Recursive Sequences

Objectives:

2. Use recursion formulas to find subsequent terms.

1. Find particular terms of sequence from the given general term.

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 TermsLet’s look at the following sequence

1, 4, 9, 16, 25, 36, 49, …,

1a 2a 3a 4a 6a5a7a

The letter a with a subscript is used to represent function values of a sequence.

The subscripts identify the location of a term.

Do you know what the rule is for the sequence? n²

How to read the subscripts:

na1na 1na

a term in the

sequence

the priorterm

the next term

Ex. 1: Find the first four terms of the sequence:

1 5a 13 na

The first term is 5

Each term

after the first

na

13 2n na a 1 5a

+ 2is

3 times the

previous term

Plus 2

Let’s be sure we understand what is given

General Term

Continued…Ex. 1: Find the first four terms of the sequence:

1 5a

2 2 13 2a a

1 5a 13 2n na a

13 2a 3(5) 2 15 2 17

3 3 13 2a a 23 2a 3(17) 2 51 2 53

4 4 13 2a a 33 2a 3(53) 2 159 2 161

n=1

n=3

n=2

n=4

given

Start with general term for n>1

Answer = 5, 17, 53, 161

Your turn: Ex 2: Find the next four terms of the sequence.

1 3a

2 2 12a a

1 3a 12n na a

12a 2(3) 6

3 3 12a a 22a 2(6) 12

4 4 12a a 32a 2(12) 24

given

Start with general term for n>1

Answer = 3, 6, 12, 24

n=1

n=3

n=2

n=4

Try another…

1 24 2n n na a a 1 22 1a a

2 1a n=1

n=3

n=2

n=4

given1 2a

3 3 1 3 24 2a a a =

=

2 14 2a a

3 24 2a a

4 – 4 = 0

0 – 2 = -24 4 1 4 24 2a a a

Answer = 2, 1, 0, -2, -8

n=5 = -8 – 0 = -85 5 1 5 24 2a a a

given

4 34 2a a

Your turnWrite 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.

Ex. 3

7, 3, -1, -5, -9, …The common difference = -4

1

1

Arithmetic

__n na a d

a

The first term = 7

1

1

Geometric

_____n na r a

a

1 4n na a

1 7a

Ex. 4

3, 6, 12, 24, 48, …

The common ratio = 2 The first term = 3

12n na a

1 3a

Last Example…

Choose the recursive formula for the given sequence.

Answer = C

Summary:

What is a recursive sequence?

Worksheet 8.3 and quest review

Homework: