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Unit 5Chapter 9: Sequences, Series, & Probability

9.1 (part 1 – Sequences)

Learning Targets:• I can use sequence notation to write the terms of a sequence.• I can use factorial notation.

Definition of Sequence

In math, the word sequence means that a collection is listed in a specific order, so it has a first member, second member, and so on.

These “members” of the sequence are called terms.

The function’s…first term = "(1)= &'second term = "(2)= &)third term= "(3)= &+ ,-. term= "(,)= &/

Definition of Sequence

• An infinite sequence is a function whose domain is the set of positive integers.

• The function values !", !$, !%, !&, … , !(, … are the terms of the sequence.

• If the domain of the function consists of the first n positive integers only, the sequence is a finite sequence.

• In an arithmetic sequence, each term has a common difference (d).

• In a geometric sequence, each term has a common ratio (r).

Example: 1, 6, 11, 16, 21, 26, … d = 5

Example: 3, 12, 48, 192, … r = 4

Example 1:

Find the first four terms of the sequences given by:

a) !" = 3% − 2!( = 3(1) − 21st term:

2nd term:

3rd term:

4th term:

!, = 3(2) − 2

!- = 3(3) − 2

!. = 3(4) − 2

= 1

= 4

= 7

= 10

Is this sequence arithmetic, geometric, or neither?

arithmetic: the common difference is 3

Example 1:

Find the first four terms of the sequences given by:

b) !" = 3 + (−1)"

1st term:

2nd term:

3rd term:

4th term:

!* = 3 + (−1)*

!+ = 3 + (−1)+

!, = 3 + (−1),

!- = 3 + (−1)-

= 2

= 4

= 2

= 4

Is this sequence arithmetic, geometric, or neither?

neither

Example 2:

Find the first five terms of the sequence given by !" = (%&)(

)"%&

!& =(−1)&2(1) − 1

!) =(−1))2(2) − 1

= −1

!- =(−1)-2(3) − 1

= 13

= −15

!0 =(−1)02(4) − 1

!2 =(−1)22(5) − 1

= 17

= −19

Explicit Formula

Because listing out just the first few terms of a sequence isn’t enough to define the sequence, a formula must be provided.

An explicit formula allows you to quickly find the nth term of the sequence without knowing any of the terms.

Examples:

!" = 3% − 2 !" = 3 + (−1)" !" = (,-).

/",-

Example 3:

Write an explicit formula for the nth term of the sequence:

a) 1, 3, 5, 7, …

':terms:

pattern:

)* =

1 2 3 4 . . . '

1 3 5 7 . . . )*Each term is 1 less than twice '

2' − 1

Example 3:

Write an explicit formula for the nth term of the sequence:

b) 2,−5, 10, −17,…

):terms:

pattern:

+, =

1 2 3 4 . . . )

2 -5 10 -17 . . . +,The terms have alternating signs whit the even positions being negative.Each term is 1 more than the square of ).

(−1),01 )2 + 1

Recursive Formula

A recursive formula requires you to be given one or more of the first few terms.

All other terms of the sequence are defined using previous terms.

!"# = %&'() )*'+", = ",-# +")ℎ ()/%%

Let’s do a couple together:

1) −17.3, −20.1, −22.9, −25.7, …

a) arithmetic, geometric, or neither?a) arithmetic, geometric, or neither?

c) recursive formula:

b) explicit formula:

,- =−2.80 −14.5

2,3 =,- =−17.3,-43 − 2.8

Let’s do a couple together:

2) 2,−12, 72, −432,…

a) arithmetic, geometric, or neither?a) arithmetic, geometric, or neither?

c) recursive formula:

b) explicit formula:

)* = −6*-.2 ∗

0). =)* =2)*-. ∗ −6

Now you try…

01c%20-%20Sequences.pdf

The Fibonacci Sequence!" = 1!% = 1!& = 2!( = 3!* = 5!, = 8

= !. +!"= !" +!%= !% +!&= !& +!(

!" = 1, !% = 1, !1 = !12% + !12", where 3 ≥ 3

5!" = 1!% = 1!1 = !12% + !12", where 3 ≥ 3

Example 4:

!"# = 14"' = "'(# − 5,where 0 ≥ 2

Write the recursive formula for the sequence: 14, 9, 4, −1,−6,…

Example 5:

!"# = 4"& = "&'# × 2,where / ≥ 2

Write the recursive formula for the sequence: 4,−8, 16, −32, 64, …

Example 6:!"# = −3"' = "'(# × 3 where . ≥ 2Find the first four terms for the sequence:

−3,−9,−27,−81

Example 7:!"# = −3"' = "'(# + * where * ≥ 2Find the first four terms for the sequence:

−3,−1, 2, 6

Factorials

If ! is a positive integer, ! factorial is defined as: !! = 1 ∗ 2 ∗ 3 ∗ 4 ∗ . . . ∗ !

As a special case, zero factorial is defined as 0! = 1

Note that

2!! = 2 !! = 2(1 ∗ 2 ∗ 3 ∗ . . . ∗ !)

(2!)! = (1 ∗ 2 ∗ 3 ∗ . . . ∗ 2!)

Example 8Find & plot the first 5 terms of the sequence given by !" = $

(&'()

("*+)!

!+ = $(('()

(+*+)!

!$ = $(-'()

($*+)!

!. = $(/'()

(.*+)!

!0 = $(1'()

(0*+)!

!2 = $(3'()

(2*+)!

= $4

5!

= $(

+!

= $-

$!

= $/

.!

= $1

0!

= ++ = 1

= $+ = 2

= 0$ = 2

= 89 =0.

= +9$0 =$.

Example 9Evaluate each factorial expression without a calculator:

a)

b)

c)

8!2! ∗ 6!

2! ∗ 6!3! ∗ 5!

(!(( − 1)!

= 8 ∗ 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 12 ∗ 1 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1=8 ∗ 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 12 ∗ 1 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 =

562 = 28

= 2 ∗ 1 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 13 ∗ 2 ∗ 1 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1=2 ∗ 1 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 13 ∗ 2 ∗ 1 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 =

63 = 2

= ( ∗ ( − 1 ∗ ( − 2 ∗ ( − 3 ∗ . . .( − 1 ∗ ( − 2 ∗ ( − 3 ∗ . . .=( ∗ ( − 1 ∗ ( − 2 ∗ ( − 3 ∗ . . .( − 1 ∗ ( − 2 ∗ ( − 3 ∗ . . . = (

Practice Problems

Page 621-622

#2, 4, 10, 14, 18, 24, 37-40, 56, 58, 60, 64, 66, 68