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01 - 9.1 Notes · 2019-03-15 · Chapter 9: Sequences, Series, & Probability. 9.1 (part 1 –Sequences) Learning Targets: •I can use sequence notation to write the terms of a sequence

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Text of 01 - 9.1 Notes · 2019-03-15 · Chapter 9: Sequences, Series, & Probability. 9.1 (part 1...

  • 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