Download Arithmetic Sequences

Arithmetic Sequences and Recursive Formulas
Why is a sequence a function and how can you model a function for an
arithmetic sequence?
F.IF.2, F.IF.3, F.BF.1, F.BF.2, F.LE.2
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Sequence – an ordered list of numbers or
items. Each element in the sequence is called
a term.
Terms can be paired with a position number
(n) – this creates a function where the domain
is set by the n’s and the range is set by the
terms. The position numbers (n) are
consecutive integers that usually start with 0
or 1.
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Ex 1 – Given a sequence f (2, 5, 8, 11, 14…),
and the domain (position) starts at 0, what is
f(4)? Explain what it represents and how you
determined your answer.
How would your answers change if the
domain started with 1?
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Ex 2 – Predict the next term in the sequence:
50, 42, 34, 26, 18,… Explain your reasoning.
Let’s discuss an example of a sequence found
in our everyday lives. What makes our
example a sequence?
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Some sequences can be described using an
algebraic model. An explicit rule allows you
to find the n-th term of the function.
Ex 3 – Find the first 4 terms of a sequence
that follows the explicit rule f(n) = 3n + 2,
assuming the domain starts with n = 1. What
do you notice about the terms and how is it
related to the equation?
◦ What would the 10th term be?
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Sometimes we can find subsequent terms in a sequence
by knowing what the previous terms are - we use
recursive rules to find the n-th term based on previous
terms.
Ex 4 – Given that f(1) = 3 and f(n) = f(n-1) + 2 for n≥2,
find the first 4 terms.
n
f(n-1) + 2
f(n)
1
Given
3
2
3
4
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Describe how you would find the 12th term?
How would you find the 50th term? Is there a more
effective way than the recursive model?
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If the difference between each term is consistent,
the function is an arithmetic sequence, where d is
the common difference.
Ex 5 – The table shows end-of-month bank
balances for an account that does not earn interest.
Month
n
1
2
3
4
5
Balanc
e
f(n)
60
80
100
120
140
a) What is the common difference? Write the
recursive rule, given f(1) = 60.
b) Write the explicit rule for the above arithmetic
sequence.
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b) Making a table may help with the explicit form.
n
f(n)
1
60 + 20 (1-1) = 60 + 20(0) = 60
2
3
4
5
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The common difference is related to something else you
are familiar with – what?
Ex 6 – If the only things you know about a particular
arithmetic sequence is that the common difference is 4
and the 3rd term is 10, how can you find the 51st term?
One way to find a rule that will work for any all
arithmetic sequences, is to look at a specific
sequence then make generalities. Let’s use 6, 9,
12, 15, 18, …
 Specific
General/Algebra
 Common difference
d=
d
 Recursive Rule
Given f(1) = 6
Given f(1)
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f(n) = f(n -1) +
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Explicit Rule
f(n) =
+
(n-1)
for n≥2 f(n) = f(n-1) + for n≥2
f(n) =
+
(n-1)
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Ex 7 – a) If f(1)= 4 and d = 10, how can you
find the 7th term of the sequence?
b) What information do you need to know in
order to find the 11th term of an arithmetic
sequence using a recursive model?
c) With a partner, discuss a real life situation
when finding terms of an arithmetic sequence
would be useful.
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Ex 8 – The graph shows how the cost of a
snowboarding trip depends on the number of
boarders.
Make a chart of the data.
Axis Title
Cost ($)
200
n
150
1
100
2
50
Cost ($)
0
0
5
Axis Title
3
4
f(n)
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a) Is the sequence arithmetic? Why or why
not?
b) Write the explicit rule.
c) Using the graph on the last slide, write an
equation of line using slope-intercept form.
d) What do you notice about the answers in b
and c? Explain. (use m and d in your
response).
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With your partner, come up with the pattern
for the famous Fibonacci Sequence (hint, it is
NOT an arithmetic sequence, but you can still
find an answer).
1, 1, 2, 3, 5, 8, 13, …
Let’s develop the recursive rule together: