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…
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*+)!
= $45!
= $(+!
= $-$!
= $/.!
= $10!
= ++ = 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