Section 9.1 – Sequences. Sequence First Term Second Term n th Term Generally, we will concentrate...

Section 9.1 – Sequences

Sequence

A sequence is a list of numbers written in an explicit order.

First Term

Second Term

nth Term

Generally, we will concentrate on infinite sequences, that is, sequences with domains that are infinite subsets of the positive integers.

Recursive FormulaA formula that requires the previous

term(s) in order to find the value of the next term.

Example: Find a Recursive Formula for the sequence below.

2, 4, 8, 16, …

Explicit FormulaA formula that requires the number of the

term in order to find the value of the next term.

Example: Find an Explicit Formula for the sequence below.

2, 4, 8, 16, …

2 n

na The Explicit

Formula is also known as the General or nth Term equation.

A sequence which has a constant difference between terms. The rule is linear.

Example: 1, 4, 7, 10, 13,…

(generator is +3)

Arithmetic Sequences

n a(n)

1 1

2 4

3 7

4 10

5 13

+3

+3

+3

+3

Discrete

3 2na n 1

1

1

3n n

a

a a

Explicit Formula Recursive Formula

Write an equation for the nth term of the sequence:

a(0) is not in the sequence! Do not include it in tables or graphs!

White Board Challenge

36, 32, 28, 24, … n=1 n=2 n=3 n=4

40, n=0

4 40na n

– 4

Seq

uenc

es ty

pica

lly s

tart

with

n=

1

First find the generator

Then find the n=0 term.

A sequence which has a constant ratio between terms. The rule is exponential.

Example: 4, 8, 16, 32, 64, …

(generator is x2)

Geometric Sequences

n t(n)

1 4

2 8

3 16

4 32

5 64

x2

x2

x2

x2

Discrete

0 1 2 3 4 5 6

2 2n

na 1

1

4

2n n

a

a a

Explicit Formula Recursive Formula

Write an equation for the nth term of the sequence:

White Board Challenge

n=1 n=2 n=3 n=43,5

n=0

35

5n

na

3, 15, 75, 375, …x5

a(0) is not in the sequence! Do not include it in tables or graphs!S

eque

nces

typi

cally

sta

rt w

ith n

=1

First find the generator

Then find the n=0 term.

New SequencesThe previous sequences were the only ones taught in Algebra 2. But, it is possible for a sequence to be neither arithmetic nor geometric.

Example: Find a formula for the general term of the sequence below

n=1 n=2 n=3 n=4 n=5

2n5n 1

1n

White Board ChallengeExample: Find a formula for the general term of the sequence below

n=1 n=2 n=3 n=4 n=5

na 12 1n

Monotonic SequenceA sequence is monotonic if it is either increasing (if for all ) or decreasing (if for all ).

Example 1: Find the first 4 terms of to see how the sequence is monotonic.

11 1 ,

22 1 ,

33 1 ,

44 1

31 2 42 3 4 5, , ,

Example 2Prove the sequence is decreasing.

If the sequence is decreasing, for all .

IF: 3

5na n

THEN: 1

3

1 5na n

3

6n

Since the denominator is smaller:

3 3

5 6n n

OR 1n na a

Therefore, is decreasing.

Bounded Sequence

A sequence is bounded above if there is a number such that

for all

A sequence is bounded below if there is a number such that

for all

If it is bounded above and below, then is a bounded sequence.

ExampleDetermine if the sequences below bounded below, bounded above, or bounded.

1.

2.

Since : 1

Since : The sequence is not bounded above.

Therefore, is bounded below.

Since : >0

Since : <1.Therefore, is

bounded.

Limit of a Sequence

A sequence has the limit and we write:

or as

if we can make the terms as close to as we like by taking sufficiently large.

If exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).

Reminder: Properties of LimitsLet b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

Constant Function

Limit of x

Limit of a Power of x

Scalar Multiple

lim ( )x cf x L

lim ( )

x cg x K

limx cb b

limx cx c

lim n n

x cx c

lim ( )x c

b f x b L

Reminder: Properties of LimitsLet b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

Sum/Difference

Product

Quotient

Power

lim ( )x cf x L

lim ( )

x cg x K

lim ( ) ( )x c

f x g x L K

lim ( ) ( )x c

f x g x L K

( )lim , 0

( )x c

f x LK

g x K

lim ( )n n

x cf x L

ExampleDetermine if the sequences below converge or diverge. If the sequence converges, find its limit.

1.

2.

3.

1lim nnn

. .11lim

L H

n 1 Converges

to 1

10lim n

nn 1/212

. .1

10lim

L H

nn

lim 2 10n

n

Diverges

lnlim nnn

1. .

1lim nL H

n 0 Converges

to 0

White Board ChallengeDetermine whether the sequence converges or diverges. If it converges, find its limit.

1Converges to 𝑎𝑛=

(𝑛+1)2−2(𝑛+1)2

Absolute Value Theorem

It is not always possible to easily find the limit of a sequence. Consider:

The Absolute Value Theorem states:

If , then .

ExampleDetermine if the sequences below converge or diverge.

1.

2.

1limn

nn

1lim nn 0

Because of the Absolute Value Theorem, Converges to 0

lim 1n

n lim1

n 1

Since the limit does not equal 0, we can not apply the Absolute Value Theorem. It

does not mean it diverges. Another test isneeded.

The sequence diverges since it does not have a limit: -1,1,-1,1,-1,…

Theorem: Bounded, Monotonic Sequences

Every bounded, monotonic sequence is convergent.

Example: Investigate the sequence below.

1 2a

2 4a

3 5a

4 5.5a

5 5.75a

5 5.85a

The sequence appears to be monotonic: It is increasing.

The sequence appears to be bounded:

The limit of the sequence appears to be 6.

Since the sequence

appears to be monotonic and

bounded, it appears to

converge to 6.