Lesson%204-4:%20Arithmetic%20and%20Geometric%20Sequences

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Lesson 4-4 Arithmetic and Geometric Sequences

• Advanced Math Topics

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Arithmetic Sequence

• Definition
• Sequence in which the difference between any term
  and the term before it is a constant.
• i.e. you are adding or subtracting the same
  number each time

• Example
• 2, 4, 6, 8,
• You are adding 2 to each term to get the next
• 10, 9, 8, 7,
• You are subtracting 1 from each term to get the
  next

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Common Difference

• Definition
• The value of an an-1

• Example
• 2, 4, 6, 8,
• 4 2 2
• 6 4 2
• 8 6 2
• d 2

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Geometric Sequence

• Definition
• Sequence in which the ratio of any term to the
  term before it is a constant.
• i.e. each term is multiplied or divided by the
  same number

• Example
• 2, 4, 8, 16,
• You multiply the previous term by 2 to get the
  next number
• 81, 27, 9, 3,
• You divide each term by 3 to get the next number

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Common Ratio

• The value of an divided by an-1

• Example
• 2, 4, 8, 16,
• 4/2 2
• 8/4 2
• 16/8 2
• So common ratio r 2

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Arithmetic sequence 10, 7, 4, 1,

• Explicit Formula
• ana1 (n-1)d
• Pattern is minus 3 from previous term so d -3
• ana1 (n-1)d
• an10 (n-1)-3
• an10 -3n 3
• an13 3n

• Recursive Formula
• anan-1 d
• Pattern is still minus 3 from previous term so d
  -3
• anan-1 d
• a110

anan-1 - 3
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Geometric Sequence 0.05, 0.25, 1.25, 6.25,

• Explicit Formula
• ana1rn-1
• Pattern is previous term times 5 so r 5
• ana1rn-1

• Recursive Formula
• an(an-1)r
• Pattern is previous term times 5 so r 5
• an(an-1)r

an.05 (5)n-1
an (an-1)5
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You decide

• Is the sequence Arithmetic, geometric, or
  neither, justify your answer.
• 100, 20, 4, 0.8,
• 1, 3, 6, 10,
• 12, 17, 22, 27,
• 0, 1, -1, 0, 1, -1,

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Write the recursive and explicit formulas for
each sequence.

• 25, 22, 19, 16,