Describing Number and Geometric Patterns
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Describing Number and Geometric Patterns
Objectives:• Use inductive reasoning in continuing patterns• Find the next term in an Arithmetic and Geometric sequence
Inductive reasoning: • make conclusions based on patterns you observe
Conjecture: • conclusion reached by inductive reasoning based on evidence
Geometric Pattern:• arrangement of geometric figures that repeat
Arithmetic Sequence• Formed by adding a fixed number to a previous term
Geometric Sequence• Formed by multiplying by a fixed number to a previous term
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• Arrangement of geometric figures that repeat• Use inductive reasoning and make conjecture as to the next figure in a pattern
Geometric Patterns
Use inductive reasoning to find the next two figures in the pattern.
Use inductive reasoning to find the next two figures in the pattern.
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Describe the figure that goes in the missing boxes.
Geometric Patterns
Describe the next three figures in the pattern below.
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Numerical Sequences and Patterns
Arithmetic Sequence
Add a fixed number to the previous termFind the common difference between the previous & next term
Find the next 3 terms in the arithmetic sequence.
2, 5, 8, 11, ___, ___, ___
+3 +3 +3 +3
14
+3
17
+3
21
What is the common difference between the first and second term?
Does the same difference hold for the next two terms?
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Arithmetic Sequence
17, 13, 9, 5, ___, ___, ___
What are the next 3 terms in the arithmetic sequence?
1 -3 -7
An arithmetic sequence can be modeled using a function rule.
What is the common difference of the terms in the preceding problem?
-4
Let n = the term number Let A(n) = the value of the nth term in the sequence
Term # 1 2 3 4 n
Term 17 13 9 5
A(1) = 17A(2) = 17 + (-4)A(3) = 17 + (-4) + (-4)A(4) = 17 + (-4) + (-4) + (-4)
Relate
Formula A(n) = 17 + (n – 1)(-4)
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Arithmetic Sequence Rule
nth term
firstterm
termnumber
Commondifference
Find the first, fifth, and tenth term of the sequence: A(n) = 2 + (n - 1)(3)
A(n) = 2 + (n - 1)(3)
First Term
A(1) = 2 + (1 - 1)(3)
= 2 + (0)(3)
= 2
A(n) = 2 + (n - 1)(3)
Fifth Term
A(5) = 2 + (5 - 1)(3)
= 2 + (4)(3)
= 14
A(n) = 2 + (n - 1)(3)
Tenth Term
A(10) = 2 + (10 - 1)(3)
= 2 + (9)(3)
= 29
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In 1995, first class postage rates were raised to 32 cents for the first ounce and 23 cents for each additional ounce. Write a function rule to model the situation.
Weight (oz) A(1) A(2) A(3) n
Postage (cents)
Real-world and Arithmetic Sequence
What is the function rule?
.32 + 23 .32+.23+.23 .32+.23+.23+.23
A(n) = .32 + (n – 1)(.23)
What is the cost to mail a 10 ounce letter?
A(10) = .32 + (10 – 1)(.23) = .32 + (9)(.23) = 2.39The cost is $2.39.
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3, 12, 48, 192, ___, _____, ______12,288
Numerical Sequences and Patterns
Geometric Sequence
• Multiply by a fixed number to the previous term• The fixed number is the common ratio
Find the common ratio and the next 3 terms in the sequence.
x 4 x 4 x 4 x 4
768
x 4
3072
x 4What is the common RATIO between the first and second term?
Does the same RATIO hold for the next two terms?
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Geometric Sequence
80, 20, 5, , ___, ___
What are the next 2 terms in the geometric sequence?
An geometric sequence can be modeled using a function rule.
What is the common ratio of the terms in the preceding problem?
Let n = the term number Let A(n) = the value of the nth term in the sequence
Term # 1 2 3 4 n
Term 80 20 5
A(1) = 80A(2) = 80 · (¼)A(3) = 80 · (¼) · (¼)
A(4) = 80 · (¼) · (¼) · (¼)
Relate
Formula A(n) = 80 · (¼)n-1
4
516
5
64
5
4
1
4
5
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Geometric Sequence Rule
nth term
firstterm
commonratio
Term number
Find the first, fifth, and tenth term of the sequence: A(n) = 2 · 3n - 1
A(n) = 2· 3n - 1
First Term
A(n) = 2 · 3n - 1
Fifth Term
A(n) = 2· 3n - 1
Tenth Term
A(1) = 2· 31 - 1 A(5) = 2 · 35 - 1 A(10) = 2· 310 - 1
A(1) = 2 A(5) = 162 A(10) = 39,366
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Write a Function Rule
Real-world and Geometric Sequence
You drop a rubber ball from a height of 100 cm and it bounces back to lower and lower heights. Each curved path has 80% of the height of the previous path. Write a function rule to model the problem.
A(n) = a· r n - 1
A(n) = 100 · .8 n - 1
What height will the ball reach at the top of the 5th path?
A(n) = 100 · .8 n - 1
A(5) = 100 · .8 5 - 1
A(5) = 40.96 cm