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Sequences. Section 14.1. Sequences. Suppose that a town’s present population of 100,000 is growing by 5% each year. After the first year, the town’s population will be: 100,000 + 0.05( 100,000 ) = 105,000 After the second year, the town’s population will be:

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Sequences

Sequences

Section 14.1


Sequences1
Sequences

Suppose that a town’s present population of 100,000 is growing by 5% each year.

After the first year, the town’s population will be:

100,000 + 0.05(100,000) = 105,000

After the second year, the town’s population will be:

105,000 + 0.05(105,000) = 110,250

After the third year, the town’s population will be:

110,250+ 0.05(110,250) ≈ 115,763


Sequences2
Sequences

If we continue to calculate, the town’s yearly population can be written as the infinite sequence of numbers

105,000, 110,250, 115,763, ….

If we decide to stop calculating after a certain year (say, the fourth year), we obtain the finite sequence

105,000, 110,250, 115,763, 121,551


Sequences3
Sequences

  • An infinite sequence is a function whose domain is the set of natural numbers {1, 2, 3, 4, …}

    • Ex. 2, 4, 6, 8, …

  • A finite sequence is a function whose domain is the set of natural numbers {1, 2, 3, 4, …, n}, where n is some natural number.

    • Ex. 1, -2, 3, -4, 5


  • W riting the terms of a sequence
    Writing the terms of a Sequence

    is known as the general term


    W riting the terms of a sequence1
    Writing the terms of a Sequence

    First term

    Second term

    Third term

    Tenth term


    Example 1
    Example 1:

    Write the first three terms of the sequence whose general term is given by .

    Evaluate , wheren is 1, 2, and 3.

    Replace n with 1


    Example 11
    Example 1:

    Write the first three terms of the sequence whose general term is given by .

    Evaluate , wheren is 1, 2, and 3.

    Replace n with 2


    Example 12
    Example 1:

    Write the first three terms of the sequence whose general term is given by .

    Evaluate , wheren is 1, 2, and 3.

    Replace n with 3


    OYO:

    • Write the first four terms of the sequence whose general term is given by .


    Example 2
    Example 2:

    If the general term of a sequence is given by , find

    a. the first term of the sequence


    Example 21
    Example 2:

    If the general term of a sequence is given by , find

    b.


    Example 22
    Example 2:

    If the general term of a sequence is given by , find

    c. the one-hundredth term of the sequence


    Example 23
    Example 2:

    If the general term of a sequence is given by , find

    d.


    OYO:

    • If the general term of a sequence is given by , find

      a. the first term of the sequence

      b.

      c. The thirtieth term of the sequence

      d.


    Example 3 finding the general term of a sequence
    Example 3: Finding the general term of a sequence

    Find a general term of the sequence whose first few terms are given.

    a. 1, 4, 9, 16, …

    Terms are all squares.


    Example 3 finding the general term of a sequence1
    Example 3: Finding the general term of a sequence

    Find a general term of the sequence whose first few terms are given.

    b.

    Terms are all reciprocals.


    Example 3 finding the general term of a sequence2
    Example 3: Finding the general term of a sequence

    Find a general term of the sequence whose first few terms are given.

    c.


    Example 3 finding the general term of a sequence3
    Example 3: Finding the general term of a sequence

    Find a general term of the sequence whose first few terms are given.

    d.

    Terms double each time.


    OYO:

    Find a general term of the sequence whose first few terms are given.

    a. b. 3, 9, 27, 81, ...


    Example 4 application
    Example 4: Application

    The amount of weight, in pounds, a puppy gains in each month of its first year is modeled by a sequence whose general term is , where nis the number of the month. Write the first five terms of the sequence, and find how much weight the puppy should gain in its fifth month.

    The puppy should gain 9 pounds in its fifth month.


    OYO:

    The value v, in dollars, of an office copier depreciates according to the sequence , where n is the time in years. Find the value of the copier after three years.

    The copier would only be worth $2022.40 after three years.


    Homework
    HOMEWORK

    • Unit 19 homework page (on the back of your unit plan)

      # 1 – 5


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