Solving Problems by Inductive Reasoning
MATH 100 Survey of Mathematical Ideas
J. Robert Buchanan
Department of Mathematics
Fall 2014
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Inductive Reasoning (1 of 2)
If we observe several similar problems and see that the
same method can be used to solve each problem, then we
have reason to believe (conjecture) that the method may
continue to be used to solve similar problems in the future.
If we encounter even one example (the counterexample)
for which the conjecture does not work, then the conjecture
is false.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Inductive Reasoning (2 of 2)
Definition
Inductive reasoning is characterized by drawing a general
conclusion (making a conjecture) from repeated observations
of specific examples. The conjecture may or may not be true.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Inductive Reasoning (2 of 2)
Definition
Inductive reasoning is characterized by drawing a general
conclusion (making a conjecture) from repeated observations
of specific examples. The conjecture may or may not be true.
Remark: inductive reasoning does not guarantee a true result,
it only provides a means of making a conjecture.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Deductive Reasoning
We may be able to establish the truth of a conjecture if we can
formally prove its absolute truth from basic principles known (or
accepted) to be true.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Deductive Reasoning
We may be able to establish the truth of a conjecture if we can
formally prove its absolute truth from basic principles known (or
accepted) to be true.
Definition
Deductive reasoning is characterized by applying general
principles to specific examples.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Deductive Reasoning
We may be able to establish the truth of a conjecture if we can
formally prove its absolute truth from basic principles known (or
accepted) to be true.
Definition
Deductive reasoning is characterized by applying general
principles to specific examples.
Note: inductive reasoning moves from specific observations to
general principles while deductive reasoning moves from
general principles to specific examples.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Determine whether the reasoning in the arguments below are
examples of deductive or inductive reasoning. Select “A” for
deductive and “B” for inductive reasoning.
If you take your medicine, you’ll feel a lot better. You take
your medicine. Therefore, you’ll feel a lot better.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Determine whether the reasoning in the arguments below are
examples of deductive or inductive reasoning. Select “A” for
deductive and “B” for inductive reasoning.
If you take your medicine, you’ll feel a lot better. You take
your medicine. Therefore, you’ll feel a lot better.
Marin’s first three children were boys. If she has another
baby, it will be a boy.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Determine whether the reasoning in the arguments below are
examples of deductive or inductive reasoning. Select “A” for
deductive and “B” for inductive reasoning.
If you take your medicine, you’ll feel a lot better. You take
your medicine. Therefore, you’ll feel a lot better.
Marin’s first three children were boys. If she has another
baby, it will be a boy.
All men are mortal. Socrates is a man. Therefore,
Socrates is mortal.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Reasoning
A premise is an assumption, law, rule, widely held idea, or
observation.
From premises we reason inductively or deductively to
obtain a conclusion.
Premises and conclusions make up a logical argument.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Identify each premise and conclusion in each of the following
argument.
1
If you take your medicine, you’ll feel a lot better. You take
your medicine. Therefore, you’ll feel a lot better.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Identify each premise and conclusion in each of the following
argument.
1
If you take your medicine, you’ll feel a lot better. You take
your medicine. Therefore, you’ll feel a lot better.
2
Marin’s first three children were boys. If she has another
baby, it will be a boy.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Identify each premise and conclusion in each of the following
argument.
1
If you take your medicine, you’ll feel a lot better. You take
your medicine. Therefore, you’ll feel a lot better.
2
Marin’s first three children were boys. If she has another
baby, it will be a boy.
3
All men are mortal. Socrates is a man. Therefore,
Socrates is mortal.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Predicting Numbers in a Sequence
We can use inductive reasoning to determine the probable next
number in the sequence below.
1 3 5 7 9
, , , , ,
3 5 7 9 11
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Predicting Numbers in a Sequence
We can use inductive reasoning to determine the probable next
number in the sequence below.
1 3 5 7 9 11
, , , , ,
3 5 7 9 11 13
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Use inductive reasoning to determine the probable next number
in each sequence below. Enter your responses as numeric
answers.
13, 18, 23, 28, 33, . . .
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Use inductive reasoning to determine the probable next number
in each sequence below. Enter your responses as numeric
answers.
13, 18, 23, 28, 33, . . .
32, 16, 8, 4, 2, . . .
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Use inductive reasoning to determine the probable next number
in each sequence below. Enter your responses as numeric
answers.
13, 18, 23, 28, 33, . . .
32, 16, 8, 4, 2, . . .
1, 4, 9, 16, 25, . . .
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (1 of 2)
Carl Friedrich Gauss was a very precocious mathematician. At
the age of 6 he determined a very simple method for adding a
list of consecutive numbers.
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (1 of 2)
Carl Friedrich Gauss was a very precocious mathematician. At
the age of 6 he determined a very simple method for adding a
list of consecutive numbers.
Suppose we wish to find the sum:
1 + 2 + 3 + · · · + 9 + 10 = S
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (1 of 2)
Carl Friedrich Gauss was a very precocious mathematician. At
the age of 6 he determined a very simple method for adding a
list of consecutive numbers.
Suppose we wish to find the sum:
1 + 2 + 3 + · · · + 9 + 10 = S
10 + 9 + 8 + · · · + 2 + 1 = S
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (1 of 2)
Carl Friedrich Gauss was a very precocious mathematician. At
the age of 6 he determined a very simple method for adding a
list of consecutive numbers.
Suppose we wish to find the sum:
1 + 2 + 3 + · · · + 9 + 10 = S
10 + 9 + 8 + · · · + 2 + 1 = S
11 + 11 + 11 + · · · + 11 + 11 = 2S
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (1 of 2)
Carl Friedrich Gauss was a very precocious mathematician. At
the age of 6 he determined a very simple method for adding a
list of consecutive numbers.
Suppose we wish to find the sum:
1 + 2 + 3 + · · · + 9 + 10 = S
10 + 9 + 8 + · · · + 2 + 1 = S
11 + 11 + 11 + · · · + 11 + 11 = 2S
10(11) = 2S
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (1 of 2)
Carl Friedrich Gauss was a very precocious mathematician. At
the age of 6 he determined a very simple method for adding a
list of consecutive numbers.
Suppose we wish to find the sum:
1 + 2 + 3 + · · · + 9 + 10 = S
10 + 9 + 8 + · · · + 2 + 1 = S
11 + 11 + 11 + · · · + 11 + 11 = 2S
10(11) = 2S
2S = 10(10 + 1)
10(10 + 1)
= 55
S =
2
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (2 of 2)
Would this procedure work if we had to sum up
{1, 2, 3, . . . , 75}?
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (2 of 2)
Would this procedure work if we had to sum up
{1, 2, 3, . . . , 75}?
1 + 2 + 3 + · · · + 74 + 75 = S
75 + 74 + 73 + · · · + 2 + 1 = S
76 + 76 + 76 + · · · + 76 + 76 = 2S
75(76) = 2S
2S = 75(75 + 1)
75(75 + 1)
S =
= 2850
2
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (2 of 2)
Would this procedure work if we had to sum up
{1, 2, 3, . . . , 75}?
1 + 2 + 3 + · · · + 74 + 75 = S
75 + 74 + 73 + · · · + 2 + 1 = S
76 + 76 + 76 + · · · + 76 + 76 = 2S
75(76) = 2S
2S = 75(75 + 1)
75(75 + 1)
S =
= 2850
2
Question: can you use inductive reasoning to find a formula for
1 + 2 + 3 + · · · + N?
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Method of Gauss (2 of 2)
Would this procedure work if we had to sum up
{1, 2, 3, . . . , 75}?
1 + 2 + 3 + · · · + 74 + 75 = S
75 + 74 + 73 + · · · + 2 + 1 = S
76 + 76 + 76 + · · · + 76 + 76 = 2S
75(76) = 2S
2S = 75(75 + 1)
75(75 + 1)
S =
= 2850
2
Question: can you use inductive reasoning to find a formula for
1 + 2 + 3 + · · · + N?
N(N + 1)
Answer: 1 + 2 + 3 + · · · + N =
.
2
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Use your i>clicker2 to submit the following sums.
1 + 2 + 3 + · · · + 100 =
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Use your i>clicker2 to submit the following sums.
1 + 2 + 3 + · · · + 100 = 5050
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Use your i>clicker2 to submit the following sums.
1 + 2 + 3 + · · · + 100 = 5050
1 + 2 + 3 + · · · + 325 =
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Use your i>clicker2 to submit the following sums.
1 + 2 + 3 + · · · + 100 = 5050
1 + 2 + 3 + · · · + 325 = 52975
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Use your i>clicker2 to submit the following sums.
1 + 2 + 3 + · · · + 100 = 5050
1 + 2 + 3 + · · · + 325 = 52975
11 + 12 + 13 + · · · + 200 =
J. Robert Buchanan
Solving Problems by Inductive Reasoning
Examples
Use your i>clicker2 to submit the following sums.
1 + 2 + 3 + · · · + 100 = 5050
1 + 2 + 3 + · · · + 325 = 52975
11 + 12 + 13 + · · · + 200 = 20045
J. Robert Buchanan
Solving Problems by Inductive Reasoning