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Patterns and Inductive Reasoning
LESSON 1-1
Additional Examples
Find a pattern for the sequence. Use the pattern to
show the next two terms in the sequence.
384, 192, 96, 48, …
Each term is half the preceding term. So the next two terms are
48 ÷ 2 = 24 and 24 ÷ 2 = 12.
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HELP
GEOMETRY
Patterns and Inductive Reasoning
LESSON 1-1
Additional Examples
Make a conjecture about the sum of the cubes of the first 25
counting numbers.
Find the first few sums. Notice that each sum is a perfect square and that the
perfect squares form a pattern.
13
13 + 23
13 + 23 + 33
13 + 23 + 33 + 43
13 + 23 + 33 + 43 + 53
=1
=9
= 36
= 100
= 225
= 12
= 32
= 62
= 102
= 152
= 12
= (1 + 2)2
= (1 + 2 + 3)2
= (1 + 2 + 3 + 4)2
= (1 + 2 + 3 + 4 + 5)2
The sum of the first two cubes equals the square of the sum of the
first two counting numbers.
HELP
GEOMETRY
Patterns and Inductive Reasoning
LESSON 1-1
Additional Examples
(continued)
The sum of the first three cubes equals the square of the sum of the
first three counting numbers.
This pattern continues for the fourth and fifth rows of the table.
13 + 23 + 33 + 43
= 100 = 102
= (1 + 2 + 3 + 4)2
13 + 23 + 33 + 43 + 53
= 225 = 152
= (1 + 2 + 3 + 4 + 5)2
So a conjecture might be that the sum of the cubes of the first 25 counting
numbers equals the square of the sum of the first 25 counting numbers,
or (1 + 2 + 3 + … + 25)2.
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HELP
GEOMETRY
Patterns and Inductive Reasoning
LESSON 1-1
Additional Examples
Find a counterexample for each conjecture.
a. A number is always greater than its reciprocal.
Sample counterexamples:
1 is not greater than 1 = 1.
1
1 is not greater than 2.
2
– 3 is not greater than – 1.
3
b. If a number is divisible by 5, then it is divisible by 10.
Sample counterexample:
25 is divisible by 5 but not by 10.
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HELP
GEOMETRY
Patterns and Inductive Reasoning
LESSON 1-1
Additional Examples
The price of overnight shipping was $8.00 in 2000, $9.50 in
2001, and $11.00 in 2002. Make a conjecture about the price in 2003.
Write the data in a table. Find a pattern.
2000
2001
2002
$8.00 $9.50 $11.00
Each year the price increased by $1.50.
A possible conjecture is that the price in 2003 will increase by $1.50.
If so, the price in 2003 would be $11.00 + $1.50 = $12.50.
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HELP
GEOMETRY