Download Inductive Reasoning and Conjecture A conjecture is an educated

Inductive Reasoning and Conjecture
A conjecture is an educated guess based on known information. Examining several
specific situations to arrive at a conjecture is called inductive reasoning. Inductive
reasoning is reasoning that uses a number of specific examples to arrive at a plausible
generalization or prediction.
Example 1 Patterns and Conjectures
The numbers represented below are called triangular numbers. Make a conjecture
about the new triangular number based on the pattern.
Observe: Each triangle is formed by adding another row of dots.
Find a Pattern: The numbers increase by 2, 3, 4, and 5.
Conjecture: The next number will increase by 6. So, it will be 15 + 6 or 21.
Example 2 Geometric Conjecture
For points P, Q, and R, PQ = 9, QR = 15, and PR = 12. Make a conjecture and draw a
figure to illustrate your conjecture.
Given: points P, Q, and R; PQ = 9, QR = 15, and PR = 12
Examine the measures of the segments. Since PQ + PR ≠ QR, the points
cannot be collinear.
Conjecture: P, Q, and R are non-collinear.
A conjecture based on several observations may be true in most circumstances, but false
in others. It takes only one false example to show that a conjecture is not true. The false
is example is called a counterexample.
Example 3 Find a Counterexample
Find a counterexample for the following statement based on the
graph.
The rates for CDs are at least 1.5% less than the rates a year ago.
Examine the graph. The statement is true for 6-month, 1-year,
and 2.5-year CDs. However, the difference in the rate for a 5-year
CD is 0.74% less, which is less than 1.5%. The statement is false
for a 5-year CD. Thus, the change in the 5-year rate is a
counterexample to the original statement.
Check for Understanding
Make a conjecture about the next item in each sequence.
1.
2. –8, –5, –2, 1, 4
Make a conjecture based on the given information. Draw a figure to
illustrate your conjecture.
suur
suur
3. PQ = RS and RS = TU
4. AB and CD intersect at P.
Determine whether each conjecture is true or false. Give a counterexample
for any false conjecture.
5. Given: x is an integer; Conjecture: –x is negative.
6. Given: WXYZ is a rectangle; Conjecture: WX = YZ and WZ = XY.
7. Most homes in the northern United States have roofs made with steep angles. In the
warmer areas of the southern states, homes often have flat roofs. Make a conjecture
about why the roofs are different.
Practice Set
Make a conjecture about the next item in each sequence.
8.
9.
10. 1, 2, 4, 8, 16, …
1 5 7
12. ,1, , ,3 , …
3 3 3
14. 2, –6, 18, –54
11. 4, 6, 9, 13, 18, …
1 1 1 1
13. 1, , , , ,...
2 4 8 16
15. –5, 25, –125, 625
Make a conjecture about the number of blocks in the next item of each
sequence.
16.
17.
Make a conjecture based on the given information. Draw a figure to
illustrate your conjecture.
18. Lines l and m are perpendicular.
19. A(–2, –11), B(2, 1), C(5, 10)
20. ∠3 and ∠4 are a linear pair.
22. P(–1, 7), Q(6, –2), R(6, 5)
24. PQRS is a rectangle.
21. RT is an angle bisector of ∠ABC.
23. HIJK is a square.
25. ∠B is a right angle in ΔABC.
Determine whether each conjecture is true or false. Give a counterexample
for any false conjecture.
26. Given: ∠1 and ∠2 are complementary angles; Conjecture: ∠1 and ∠2 form a right
angle.
27. Given: m + y ≥ 10, y ≥ 4; Conjecture: m ≤ 6
28. Given: points W, X, Y, and Z; Conjecture: W, X, Y, and Z are noncollinear.
29. Given: A(–4, 8), B(3, 8), C(3, 5); Conjecture: ΔABC is a right triangle.
30. Given: n is a real number; Conjecture: n2 is a nonnegative number.
31. Given: DE = EF; Conjecture: E is the midpoint of DF .
32. Given: JK = KL = LM = MJ; Conjecture: JKLM forms a square.
33. Given: noncollinear points R, S, and T; Conjecture: RS,ST , and RT form a
triangle.
For problems 34-36, use the following information.
Hydrocarbons are molecules composed of only carbon (C) and hydrogen (H) atoms.
The simplest hydrocarbons are called alkanes. The first three alkanes are shown below.
34. Make a conjecture about butane, which is the next compound in the group. Write
its structural formula.
35. Write the chemical formula for the 7th compound in the group.
36. Develop a rule that you could use to find the chemical formula of the nth substance
in the alkane group.
37. What is the next term in the sequence 1, 1, 2, 3, 5, 8?
a. 11
b. 12
c. 13
d. 14
38. If the average of six numbers is 18 and the average of three of the numbers is 15,
then what is the sum of the remaining three numbers?
a. 21
b. 45
c. 53
d. 63
Determine which values in the given replacement set make the inequality
true.
39. x + 2 > 5 {2, 3, 4, 5}
40. 12 – x < 0 {11, 12, 13, 14}
41. 5x + 1 > 25 {4, 5, 6, 7}