Transcript 3-7 Recursive Formulas For Arithmetic Sequences

Do Now – Complete the volume
problems in terms of pi
NO CALCULATORS
EXAMPLE
Find the volume of a cone, in terms of pi, with
a radius of 3 inches and a height of 6 inches.
V = 1/3π(32)(6)
V = 1/3(9)(6) π
V = 3(6) π
V = 18π
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Recursive Formulas
For Arithmetic
Sequences
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Do you know the definition
of “recursive”?
According to Merriam Webster:
relating to a procedure that can
repeat itself indefinitely
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Mathematically Speaking
Recursive Formula:
– Formula where each term is based on the
term before it.
Arithmetic Sequence:
– Sequence with a constant difference between
terms.
Domain of arithmetic sequences
You can use arithmetic sequences
as linear functions to model realworld situations. While the domain
of some linear functions is the set of
all real numbers, the domain of an
arithmetic sequence as a linear
function is the set of counting
numbers, {1, 2, 3, 4, …}.
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Example
•
•
Consider the sequence generated by
2000, 2040, 2080, 2120, 2160,. . .
Describe this sequence in words.
ìï a1 = 2000
í
ïî an = an-1 + 40
n = positive integers ≥ 2
How is a recursive
function represented?
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YOU TRY: Write an arithmetic sequence for
the range of this sequence,
If you buy a new car, you might be advised
to have an oil change after driving 1000
miles and every 3000 miles thereafter. Then
the following sequence gives the mileage
when oil changes are required:
1000 4000 7000 10000 13000 16000
ìï a1 = 1000
í
ïî an = an-1 + 3000
n = positive integers ≥ 2
Another example; This time write the
recursive formula
• Briana borrowed $870 from her parents for
airfare to Europe. She will pay them back
at the rate of $60.00 per month. Let an be
the amount she still owes after n months.
Find a recursive formula for this sequence.
ìï a1 = 870
í
ïî an = an-1 - 60
n = positive integers ≥ 2
Graph of an Arith. Seq.
• Discrete Domain and
Range
• Constant Increase or
Decrease
• Your first input is 1,
not 0
ìïa1 =1
í
ïîan = an-1 + 2,
n³2
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3-8
Explicit Formulas
For Arithmetic
Sequences
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Arithmetic Sequences
Explicit Formula
– Formula where any term can be found by
substituting the number of that term.
– We can develop an explicit formula for an
Arithmetic Sequence from the recursive
formula
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Explicit Formula
a1  1000

an  an1  3000; n  2
n
1
2
3
4
an
1000
1000+3000=4000
4000+3000=7000
7000+3000=10000
an  a1  (n  1)d
d
3000
3000
3000
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• So, for our oil change example, the explicit
formula looks like:
an  1000  (n  1)3000
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Examples
1. Find the 40th term of the arithmetic
sequence 100,97,94,91,…..
2. In a concert hall the 1st row has 20 seats
in it, and each subsequent row has 2
more seats than the row in front of it. If
the last row has 64 seats, how many
rows are in the concert hall?
3. Answer 23 rows!