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Transcript
```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


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
or 1.


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
domain started with 1?


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?


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?


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


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?




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.

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


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)

f(n) = f(n -1) +

Explicit Rule
f(n) =
+
(n-1)
for n≥2 f(n) = f(n-1) + for n≥2
f(n) =
+
(n-1)



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.

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)




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.