Transcript Document

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 terms
a 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
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