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SS.01.4 - Recursion Formulas

MCR3U - Santowski

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(A) Review

the formula for generating the terms of an arithmetic sequence is tn = a + (n - 1)d

the formula for generating the terms of a geometric sequence is tn = arn-1

these two formulas are referred to as explicit formulas

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(B) An Investigation Into Recursion Formulas

ex 1. A car worth $26,000 depreciates by 10% every year.

(i) Determine the dollar amount of the depreciation at the end of year 1.(ii) Determine the residual value of the car at the end of year 1. What percent of the original selling price does this residual value represent?(iii) Repeat the above steps for year 2.(iv) Write the first five terms of the sequence that represents the residual value of the car at the end of each successive year.(v) Let tn represent any term in the sequence. Write an expression for the term previous to tn.(vi) Write a formula that can be used to generate the terms of the sequence that uses the previous term to calculate any term (except the first one)

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(B) An Investigation Into Recursion Formulas

ex 2. A pair of rabbits one month old is too young to produce more rabbits. But by the end of the second month, they produce a pair of rabbits, and a pair of rabbits every month after that. Each new pair of rabbits does the same thing, producing a pair of rabbits every month, starting after the second month.

(i) Set up a chart or table to show how many rabbits there are each month until the start of the 9th month.(ii) List the numbers of pairs of rabbits in order as the first 9 terms of a sequence.(iii) Is this sequence arithmetic or geometric? Is this a sequence?(iv) Describe in words and by a formula how to generate the next 5 terms of the sequence.

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(C) Recursion Formulas

in a recursive sequence, a new term is generated from the previous term or termsa recursive formula shows how to calculate each term from the previous term(s)a recursive formula has at least one known term

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(D) Examples

ex 1. Write the first four terms of the sequence t1 = 2 and tn = 3tn-1 + 5

ex 2. Write the first four terms of the sequence t1 = -1, t2 = 1, and tn = 2tn-2 - 3tn-1

ex 3. Write a recursive formula for the sequence 1, 2, -1, 3, -4, 7, -11

ex 4. Write a recursive formula for the sequence 2, 6, 18, 54

ex 5. Write a recursive formula for the sequence 1,2,4,7,11,16

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(D) Examples

ex 6. To relieve the pain of chromic headaches, my doctor prescribes 500 mg of aspirin every 6 hours. At this rate, only 26% of the aspirin remains in my body when I take the next dose. Determine the first 6 terms of the sequence representing the total amount of aspirin in my body when I take the next dose. Write the recursive formula and graph it. What happens to the aspirin in my body as I take more and more aspirin?

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(E) Homework

AW, page 36, Q6,7,9,10,11,12

Nelson Text, p32-34, Q1-3,5,6,8,9,11,13,16,17