8.1 Sequences and Series

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8.1 Sequences and Series

• Essential Questions Think about your routine
  every morning, are there patterns? What about
  school?

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Definition of a 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.

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Finding Terms of a Sequence

• The first four terms of the sequence given by
  are

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Finding Terms of a Sequence

• Write out the first five terms of the sequence
  given by

Solution
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Finding the nth term of a Sequence

• Write an expression for the apparent nth term
  (an) of each sequence.
• a. 1, 3, 5, 7,
  b. 2, 5, 10, 17,

Solution

• n 1 2 3 4 . . . n
• terms 1 3 5 7 . . . an

• n 1 2 3 4 n
• terms 2 5 10 17 an

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Additional Example

• Write an expression for the apparent nth term of
  the sequence

Solution
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Factorial Notation

• If n is a positive integer, n factorial is
  defined by
• As a special case, zero factorial is defined
  as 0! 1.
• Here are some values of n! for the first
  several nonnegative integers. Notice that 0! is
  1 by definition.

The value of n does not have to be very large
before the value of n! becomes huge. For
instance, 10! 3,628,800.
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Finding the Terms of a Sequence Involving
Factorials

• List the first five terms of the sequence given
  by
• Begin with n 0.

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Evaluating Factorial Expressions

• Evaluate each factorial expression. Make sure
  you use parentheses when necessary.
• a. b.
  c.

Solution a. b. c.
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Have you ever seen this sequence before?

• 1, 1, 2, 3, 5, 8
• Can you find the next three terms in the
  sequence?
• Hint 13,
• 21, 34
• Can you explain this pattern?

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The Fibonacci Sequence

• Some sequences are defined recursively. To
  define a sequence recursively, you need to be
  given one or more of the first few terms. A
  well-known example is the Fibonacci Sequence.
• The Fibonacci Sequence is defined as follows

Write the first six terms of the Fibonacci
Sequence
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Summation Notation

• Definition of Summation Notation
• The sum of the first n terms of a sequence is
  represented by
• Where i is called the index of summation, n is
  the upper limit of summation and 1 is the lower
  limit of summation.

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Summation Notation for Sums

• Find each sum.
• a. b.
  c.

Solution a.
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Solutions continued

• b.
• c.

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How to Input Sums in your calculator
Be sure you are in sequence mode on the
calculator!

• Good news! This can all be done using the TI-84
  Plus graphing calculator.
• To enter in example a, hit the following keys
  
• The following screen will appear
• Now hit
  5. Then, hit

• Good news! This can all be done using the TI-84
  Plus graphing calculator.
• To enter in example a, hit the following keys
  
• The following screen will appear
• Now hit
  5. Then, hit

The following screen should appear
Choose 5.
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This is what your calculator screen should look
like
Now type in the sum, the variable, the lower
limit, the upper limit, and the increment
(default is 1).
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Assignment

• Page 587-588
• 2-32 even, 57-59 odd, 62-64 even, 66-70 even,
  80-94 even