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11.5 Recursive Rules for Sequences p. 681

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Explicit Rule A function based on a term’s position, n, in a sequence. All the rules for the nth term that we’ve been working with are explicit rules; such as a n =a 1 r n-1.

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Recursive Rule Gives the first term(s) of a sequence and an equation that relates the given term(s) to the next terms in the sequence. For example: Given a 0 =1 and a n =a n-1 -2 The 1 st five terms of this sequence would be: a 0, a 1, a 2, a 3, a 4 OR 1, -1, -3, -5, -7

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Example: Write the 1 st 5 terms of the sequence. a 1 =2, a 2 =2, a n =a n-2 -a n-1 a 3 =a 3-2 -a 3-1 =a 1 -a 2 =2-2=0 a 4 =a 4-2 -a 4-1 =a 2 -a 3 =2-0=2 a 5 =a 5-2 -a 5-1 =a 3 -a 4 =0-2=-2 2, 2, 0, 2, -2 1 st term 2 nd term

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Example: Write the indicated rule for the arithmetic sequence with a 1 =15 and d=5. Explicit rule a n =a 1 +(n-1)d a n =15+(n-1)5 a n =15+5n-5 a n =10+5n Recursive rule (*Use the idea that you get the next term by adding 5 to the previous term.) Or a n =a n-1 +5 So, a recursive rule would be a 1 =15, a n =a n-1 +5

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Example: Write the indicated rule for the geometric sequence with a 1 =4 and r=0.2. Explicit rule a n =a 1 r n-1 a n =4(0.2) n-1 Recursive rule (*Use the idea that you get the next term by multiplying the previous term by 0.2) Or a n =r*a n-1 =0.2a n-1 So, a recursive rule for the sequence would be a 1 =4, a n =0.2a n-1

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Write a recursive rule for the sequence 1,2,2,4,8,32,…. First, notice the sequence is neither arithmetic nor geometric. So, try to find the pattern. Notice each term is the product of the previous 2 terms. Or, a n-1 *a n-2 So, a recursive rule would be: a 1 =1, a 2 =2, a n = a n-1 *a n-2

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Example: Write a recursive rule for the sequence 1,1,4,10,28,76. Is the sequence arithmetic, geometric, or neither? Find the pattern. 2 times the sum of the previous 2 terms Or 2(a n-1 +a n-2 ) So the recursive rule would be: a 1 =1, a 2 =1, a n = 2(a n-1 +a n-2 )

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Assignment

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