Download Write a conjecture that describes the pattern in each sequence

meeting. The next month is in the sequence is July.
2-1 Inductive Reasoning and Conjecture
Write a conjecture that describes the pattern in
each sequence. Then use your conjecture to
find the next item in the sequence.
18. 1, 4, 9, 16
SOLUTION: 2
1=1
2
4=2
2
9=3
2
16 = 4
Each element is the square of increasing natural
numbers.
ANSWER: Each meeting is two months after the previous
meeting; July.
28. FITNESS Gabriel started training with the track
team two weeks ago. During the first week, he ran
0.5 mile at each practice. The next three weeks he
ran 0.75 mile, 1 mile, and 1.25 miles at each practice.
If he continues this pattern, how many miles will he
be running at each practice during the 7th week?
SOLUTION: Form a sequence using the given information.
Running speed at each practice: 0.5, 0.75, 1, 1.25, …
Each element in the pattern is 0.25 more than
previous element.
2
5 = 25
The next element in the sequence is 25.
ANSWER: Each element is the square of increasing natural
numbers; 25.
23. Club meetings: January, March, May, . . .
SOLUTION: March is two month after January and May is two
months after March. Each meeting is two months after the previous
meeting. The next month is in the sequence is July.
ANSWER: Each meeting is two months after the previous
meeting; July.
28. FITNESS Gabriel started training with the track
team two weeks ago. During the first week, he ran
0.5 mile at each practice. The next three weeks he
ran 0.75 mile, 1 mile, and 1.25 miles at each practice.
If he continues this pattern, how many miles will he
be running at each practice during the 7th week?
SOLUTION: Form a sequence using the given information.
Running speed at each practice: 0.5, 0.75, 1, 1.25, …
Each element in the pattern is 0.25 more than
previous element.
He will run 2 miles during the 7th week.
ANSWER: 2 mi
30. VOLUNTEERING Carrie collected canned food
for a homeless shelter in her area each day for one
week. On day one, she collected 7 cans of food. On
day two, she collected 8 cans. On day three she
collected 10 cans. On day four, she collected 13
cans. If Carrie wanted to give at least 100 cans of
food to the shelter and this pattern of can collecting
continued, did she meet her goal?
SOLUTION: First day : 7
Second day: 7 + 1 = 8
Third day: 8 + 2 = 10
Fourth day: 10 + 3 = 13
Fifth day: 13 + 4 = 17
Sixth day: 17 + 5 = 22
Seventh day: 22 + 6 = 28
7 + 8 + 10 + 13 + 17 + 22 + 28 = 105
Yes. She collected 105 can at the end of one week.
ANSWER: Yes; she collected 105 cans.
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He will run 2 miles during the 7th week.
Make a conjecture about each value or
geometric relationship.
31. the product of two odd numbers
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Yes. She collected 105 can at the end of one week.
ANSWER: 2-1 Inductive
Reasoning and Conjecture
Yes; she collected 105 cans.
Make a conjecture about each value or
geometric relationship.
31. the product of two odd numbers
SOLUTION: The product of two odd numbers is an odd number.
Examples:
ANSWER: The product is an odd number.
33. the relationship between a and c if ab = bc, b ≠ 0
SOLUTION: If
Examples:
, then a and c are equal.
ANSWER: The points equidistant from A and B form the
perpendicular bisector of
.
37. the relationship between the areas of a square with
side x and a rectangle with sides x and 2x
SOLUTION: Area of the square = x
2
2
Area of the rectangle = 2x
The area of the rectangle is two times the area of the
square.
ANSWER: The area of the rectangle is two times the area of the
square.
JUSTIFY ARGUMENTS Determine whether
each conjecture is true or false . Give a
counterexample for any false conjecture.
40. If n is a prime number, then n + 1 is not prime.
SOLUTION: False
Sample answer: If
number.
, then
, a prime
ANSWER: False; Sample answer: If n = 2, then n + 1 = 3, a
prime number.
ANSWER: They are equal.
35. the relationship between
equidistant from A and B
and the set of points
SOLUTION: The points equidistant from A and B form the
perpendicular bisector of
.
41. If x is an integer, then –x is positive.
SOLUTION: False.
Sample answer: Suppose x = 2, then −x = −2.
ANSWER: False; sample answer: Suppose x = 2, then −x = −2.
42. If
are supplementary angles, then
form a linear pair.
SOLUTION: False
ANSWER: The points equidistant from A and B form the
perpendicular bisector of
.
37. the relationship between the areas of a square with
side x and a rectangle with sides x and 2x
SOLUTION: 2
AreaManual
of the- Powered
square by
= xCognero
eSolutions
2
Area of the rectangle = 2x
The area of the rectangle is two times the area of the
Since ∠3 and ∠2 are supplementary m∠3 + m∠2 =
180. However, to be a linear pair, then need to be
adjacent angles and have noncommon side that are
opposite rays.
ANSWER: False; sample answer:
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False.
Sample answer: Suppose x = 2, then −x = −2.
This follows the Pythagorean Theorem. So the
statement is true.
ANSWER: 2-1 Inductive
Reasoning and Conjecture
False; sample answer: Suppose x = 2, then −x = −2.
42. If
are supplementary angles, then
form a linear pair.
ANSWER: true
45. If the area of a rectangle is 20 square meters, then
the length is 10 meters and the width is 2 meters.
SOLUTION: False
Sample answer: The length could be 4 m and the
width could be 5 m.
SOLUTION: False
ANSWER: False; sample answer: The length could be 4 m and
the width could be 5 m.
Since ∠3 and ∠2 are supplementary m∠3 + m∠2 =
180. However, to be a linear pair, then need to be
adjacent angles and have noncommon side that are
opposite rays.
FIGURAL NUMBERS Numbers that can be
represented by evenly spaced points arranged
to form a geometric shape are called figural
numbers. For each figural pattern below,
a. write the first four numbers that are
represented,
b. write a conjecture that describes the pattern
in the sequence,
c. explain how this numerical pattern is shown
in the sequence of figures,
d. find the next two numbers, and draw the next
two figures.
ANSWER: False; sample answer:
43. If you have three points A, B, and C, then A, B, C
are noncollinear.
SOLUTION: False
Sample answer:
47. ANSWER: False; sample answer:
44. If in
right triangle.
is a
SOLUTION: This follows the Pythagorean Theorem. So the
statement is true.
ANSWER: true
45. If the area of a rectangle is 20 square meters, then
the length is 10 meters and the width is 2 meters.
SOLUTION: False
Sample answer: The length could be 4 m and the
eSolutions
Manual
width
could- Powered
be 5 m.by Cognero
ANSWER: SOLUTION: a. Each points represents 1. Count the points for
each figure.
1, 4, 9, 16
b. A few different methods can be used here.Look
for a basic pattern first.
1, 4, 9, and 16 are all perfect squares, and the figures
all look like squares, so this could be the pattern..
Another method could be adding 3 to 1 to get the
second number, 4. Continue adding the next odd
number to the previous number to get the next
number in the sequence.
1+3=4
4+5=9
9 + 7 = 16
c. It looks like an extra row and column of points is
added to the previous square to create a larger
square. The number of points on each side of thePage 3
square is equal to the square's position in the pattern.
Each figure is the previous figure with an additional
1+3=4
4+5=9
9 + 7 = 16
2-1 Inductive
Reasoning and Conjecture
c. It looks like an extra row and column of points is
added to the previous square to create a larger
square. The number of points on each side of the
square is equal to the square's position in the pattern.
Each figure is the previous figure with an additional
row and column of points added, which is 2(position
number) – 1. One is subtracted since 2(position
number) counts the corner point twice. 2(position
number) – 1 is always an odd number.
d. Add the row and column minus one. (Add 9 points
and then 11 points)
51. GOLDBACH’S CONJECTURE Goldbach’s
conjecture states that every even number greater
than 2 can be written as the sum of two primes. For
example, 4 = 2 + 2, 6 = 3 + 3,
and 8 = 3 + 5.
a. Show that the conjecture is true for the even
numbers from 10 to 20.
b. Given the conjecture All odd numbers greater
than 2 can be written as the sum of two primes, is
the conjecture true or false ? Give a counterexample
if the conjecture is false.
SOLUTION: a. 10 = 5 + 5, 12 = 5 + 7, 14 = 7 + 7, 16 = 5 + 11, 18
= 7 + 11, and 20 = 7 + 13.
b. False; 3 cannot be written as the sum of two
primes.
ANSWER: a. 1, 4, 9, 16
b. Sample answer: Start by adding 3 to 1 to get the
second number, 4. Continue adding the next odd
number to the previous number to get the next
number in the sequence.
c. Sample answer: Each figure is the previous figure
with an additional row and column of points added,
which is 2(position number) – 1. One is subtracted
since 2(position number) counts the corner point
twice. 2(position number) – 1 is always an odd
number.
d. 25, 36
ANSWER: a. 10 = 5 + 5, 12 = 5 + 7, 14 = 7 + 7, 16 = 5 + 11, 18
= 7 + 11, 20 = 7 + 13
b. False; 3 cannot be written as the sum of two
primes.
55. ERROR ANALYSIS Juan and Jack are discussing
prime numbers. Juan states a conjecture that all
prime numbers are odd. Jack disagrees with the
conjecture and states all prime numbers are not odd.
Is either of them correct? Explain.
SOLUTION: Jack is correct.
2 is an even prime number.
ANSWER: Jack; 2 is an even prime number.
51. GOLDBACH’S CONJECTURE Goldbach’s
conjecture states that every even number greater
than 2 can be written as the sum of two primes. For
example, 4 = 2 + 2, 6 = 3 + 3,
and 8 = 3 + 5.
a. Show that the conjecture is true for the even
numbers from 10 to 20.
b. Given the conjecture All odd numbers greater
than 2 can be written as the sum of two primes, is
the conjecture true or false ? Give a counterexample
if the conjecture is false.
SOLUTION: a. 10 = 5 + 5, 12 = 5 + 7, 14 = 7 + 7, 16 = 5 + 11, 18
= 7 +Manual
11, and
20 = 7by+Cognero
13.
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b. False; 3 cannot be written as the sum of two
primes.
57. ANALYZE RELATIONSHIPS Consider the
conjecture If two points are equidistant from a
third point, then the three points are collinear. Is
the conjecture true or false ? If false, give a
counterexample.
SOLUTION: The conjecture is sometimes true.
If the two points create a straight angle that includes
the third point, then the conjecture is true. If the two points do not create a straight angle with
the third point, then the conjecture is false.
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Jack is correct.
2 is an even prime number.
ANSWER: 2-1 Inductive
Reasoning and Conjecture
Jack; 2 is an even prime number.
57. ANALYZE RELATIONSHIPS Consider the
conjecture If two points are equidistant from a
third point, then the three points are collinear. Is
the conjecture true or false ? If false, give a
counterexample.
SOLUTION: The conjecture is sometimes true.
If the two points create a straight angle that includes
the third point, then the conjecture is true. If the two points do not create a straight angle with
the third point, then the conjecture is false.
ANSWER: Sample answer: Sometimes; if the two points create
a straight angle that includes the third point, then the
conjecture is true. If the two points do not create a
straight angle with the third point, then the conjecture
is false.
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