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Patterns and Sequences

Patterns and Sequences. To identify and extend patterns in sequences To represent arithmetic sequences using function notation. Vocabulary. Sequence –an ordered list of numbers that often form a pattern Term of a sequence – each number in the sequence

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Patterns and Sequences

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  1. Patterns and Sequences To identify and extend patterns in sequences To represent arithmetic sequences using function notation

  2. Vocabulary • Sequence –an ordered list of numbers that often form a pattern • Term of a sequence – each number in the sequence • Arithmetic sequence – when the difference between each consecutive term is constant • Common difference – the difference between the terms • Geometric sequence – when consecutive terms have a common factor • Recursive formula – a formula that related each term of the sequence to the term before it • Explicit formula – a function rule that relates each term of the sequence to the term number

  3. EX1: Describe a pattern in each sequence. What are the next two terms of each sequence?3, 10, 17, 24, 31, … Explicit Formula Recursive formula A1 = first term An = A(n-1) + d A1 = 3 An = A(n-1) + 7 The patterns is to add 7 to the previous term. The next two terms are 31+7 = 38 and 38+7 = 45 • Common difference = +7 • A1 = 3 An = A1 + (n-1)d An = 3 + (n-1)7 An = 3 + 7n – 7 An = 7n -4 Arithmetic sequence

  4. EX2: Describe a pattern in each sequence. What are the next two terms of each sequence?2, -4, 8, -16, … Explicit Formula Recursive formula A1 = first term An = A(n-1) * r A1 = 2 An = A(n-1)(-2) The patterns is to multiply by -2 to the previous term. The next two terms are -16*-2 = 32 and 32*-2 = -64 • Common factor = -2 • A1 = 2 An = A1 (rn) An = 2(-2n) Geometric sequence

  5. Try: Describe a pattern in each sequence. What are the next two terms of each sequence?28, 17, 6, … Explicit Formula Recursive formula A1 = 3 An = A(n-1) + d A1 = 28 An = A(n-1) -11 The patterns is to subtract 11 to the previous term. The next two terms are 6-11= -5 and -5-11 = -16 • Common difference = -11 • A1 = 28 An = A1 + (n-1)d An = 28 + (n-1)-11 An = 28 - 11n – 11 An = -11n +14

  6. Ex3: A subway pass has a starting value of $100. After one ride, the value of the pass is $98.25, after two rides, its value is $94.75. Write an explicit formula to represent the remaining value of the pass after 15 rides? How many rides can be taken with the $100 pass? Explicit formula 15 Rides: An = 101.75 – 1.75(15) An = 101.75 – 1.75(15) = 101.75 – 26.25 = $75.50 You will have $75.50 left on the pass after riding 15 times. • An = 100 + (n-1)-1.75 • An = 100 - 1.75n + 1.75 • An = 101.75 - 1.75n $100 pass: 0 = 101.75-1.75n -101.75 -101.75 -101.75 = -1.75n -1.75 -1.75 58  n You can take about 58 bus rides for $100.

  7. Ex4: Writing an explicit function from a recursive function. An = A(n-1) + 2; A1= 32 An = A(n-1) -5 ; A1 = 21 Common difference = -5 A1 = 21 An = 21 + (n-1)-5 An = 21 – 5n + 5 An = 26 – 5n • Common difference = +2 • A1 = 32 An = 32 + (n-1)2 An = 32 + 2n-2 An = 30 + 2n

  8. Ex 5: Write a recursive formula from an explicit formula An = 76 + (n-1)10 An = 1 + (n-1)-3 Common difference = -3 An = 1 An = 1 An = A(n-1) – 3 • Common difference = 10 • An = 76 An = 76 An = A(n-1) + 10

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