Inductive and Deductive Reasoning

Inductive Reasoning
The type of reasoning that forms a conclusion based on the examination
is called inductive reasoning. The conclusion formed by using inductive
reasoning is a conjecture, since it may or may not be correct.

Inductive Reasoning

Inductive reasoning is the process of reaching a general conclusion by

examining specific examples

When you examine a list of numbers and predict the next in the list
according to some pattern you have observed, you are using inductive
reasoning.
Solution
A statement may have many counterexamples, but we need only find one
counterexample to verify that the statement is false.
a. Let x = 0. Then |0| = 0. Because 0 is not greater than 0, we have found a
counterexample. Thus “for all numbers x, |x| > 0” is a false statement.
b. For x = 1 we have 12 = 1. Since 1 is not greater than 1, we have found a
counterexample. Thus “for all numbers x, x2 > x.” is a false statement.
c. Consider x = -3. Then (−3)2 = 9 = 3. Since 3 is not equal to -3, we have
found a counterexample. Thus “for all numbers x, 𝑥 2 = x “ is a false statement.
EXAMPLE 1 Use Inductive Reasoning to Predict a Number

Use inductive reasoning to predict the next number in each of the following lists.
a. 3, 6, 9, 12, 15, ? b. 1, 3, 6, 10, 15, ?

Solution
a. Each successive number is 3 larger than the preceding number. Thus we predict that the
number in the list is 3 larger than 15, which is 18.
b. The first two numbers differ by 2. The second and the third numbers differ by 3. It appears that
the difference between any two numbers is always 1 more than the preceding difference. Since
10 and 15 differ by 5, we predict that the next number in the list will be 6 larger than 15, which
is 21.
 Inductive reasoning is not used just to predict the next number
in a list. In Example 2 we use inductive reasoning to make conjecture
about an arithmetic procedure.

EXAMPLE 2 Use Inductive Reasoning to Make a Conjecture

Consider the following procedure: Pick number. Multiply the number

by 8, add 6 to the product, divide the sum by 2, and subtract 3.
Complete the above procedure for several different numbers.
Use inductive reasoning to make a conjecture about the relationship
between the size of the resulting number and the size of original
number.
Solution
Suppose we pick 5 as our original number. Then the procedure would produce the
following results:

Original number: 5
Multiply by 8 8 x 5 = 40
Add 6: 40 + 6 = 46
Divide by 2: 46 ÷ 2 = 23
Subtract 3: 23 – 3 = 20
We started with 5 and followed the procedure to produce 20. Starting with 6 as our
original number produces a final result of 24. Starting with 10 produces a final
result of 40. Starting with 100 produces a final result of 400. In each of these cases
the resulting number is four times the original number. We conjecture that
following the given procedure a number that is four times the original number.
EXAMPLE 3 Use Inductive Reasoning to Solve an Application
Use the data in the above table and inductive reasoning to answer to answer each
of the following questions.
a. If a pendulum has a length of 49 units, what is its period?
b. If the length of a pendulum is quadrupled, what happens to its period?
Solution
a. In the table, each pendulum has a period that is the square root of its length.
Thus we conjecture that a pendulum with a length of 49 units will have a period
of 7 heartbeats.
b. In the table, pendulum with a length of 4 units. A pendulum with a length of 16
units has a period that is twice that of a pendulum with a length of 4 units. It
appears that quadrupling the length of a pendulum doubles its period.
Counterexamples
A statement is a true statement provided that it is true in all cases. If you can find
one case for which a statement is not true, called a counterexample, then the
statement by finding a counterexample for each.

EXAMPLE 4 Find a Counterexample

Verify that each of the following statements is a false statement by finding

example.
For all numbers x:
a. |x|>0 b. x2 >x c. √x2 = x
Inductive Reasoning vs. Deductive Reasoning
In example 6 we analyze arguments to determine whether they use inductive or deductive
reasoning.
EXAMPLE 6 Determine Types of Reasoning
Determine whether each of the following arguments is an example of inductive reasoning
or deductive reasoning.

a. During the past 10 years, a tree has produced plums every other year. Last year the tree
did not produce plums, so this year the tree will produce plums.
b. All home improvements cost more than the estimate. The contractor estimated that my
home improvement will cost $35,000. Thus my home improvement will cost more than
$35,000.
Solution:
a. This argument reaches a conclusion based on specific examples, so it is an example of
inductive reasoning.
b. Because the conclusion is a specific case of general assumption, this argument is an
example of deductive reasoning
EXAMPLE 6 Solve a Logic Puzzle

Each of four neighbors, Sean, Maria, Sarah and Brian, has a different occupation (editor,
banker, chef, or dentist). From the following clues, determine the occupation of each
neighbor.
1. Maria gets home from work after the banker but before the dentist.
2. Sarah, who is the last to get home from work, is not the editor.
3. The dentist and Sarah for work at the same time.
4. The banker lives next door to Brian.

Solution:
From clue 1, Maria is not the banker but before the dentist. In the following chart, write X1
(which stands for “ruled out by clue 1”) in the Banker and the Dentist columns of Maria’s
row.
 Editor Banker Chef Dentist
Sean
Maria X1 X1
Sarah
Brian

From clue 2, Sarah is not the editor. Write X2 (ruled out by clue 2) in the Editor column of
Sarah’s row. We know from clue 1 that the banker is not the last to get home and we know
from clue 2 that Sarah is the last to get home; therefore, Sarah is not the banker. Write X2
in the Banker column of Sarah’s row.
Editor Banker Chef Dentist
Sean
Maria X1 X1
Sarah X2 X2
Brian
From clue 3, Sarah is not the dentist. Write X3 for this condition. There are not Xs for three
of the four occupations in Sarah’s row; therefore, Sarah must be the chef. Place a √ in that
box. Since Sarah is the chef, none of the other three people can be the chef. Write X3 for
these conditions. There are now Xs for three of the four occupations in Maria’s row;
therefore Maria must be the editor. Insert a √ to indicate that Maria is the editor, and write
X3 twice to indicate that neither Sean nor Brian is the editor.

Editor Banker Chef Dentist

Sean X3 X3
Maria √ X1 X3 X1
Sarah X2 X2 √ X3
Brian X3 X3

From clue 4, Brian is not the banker. Write X4 for this condition. See the following table.
Since there are three Xs in the Banker column, Sean must be the banker. Place a √ in that
box. Thus Sean cannot be the dentist, Write X4 in that box. Since there are 3 Xs in the
Dentist column, Brian must be the dentist. Place a √ in that box.
 Editor Banker Chef Dentist
Sean X3 √ X3 X4
Maria √ X1 X3 X1
Sarah X2 X2 √ X3
Brian X3 X4 X3 √

Sean is the banker, Maria is the editor, Sarah is the chef, and Brian is the dentist.

EXAMPLE 1 Predict the Next Term of a Sequence

Use a difference table to predict the next term in the sequence.
2,7,24,59,118,207,….
Solution
Construct a difference table as shown below.
Sequence 2 7 24 59 118 207 332

First differences 5 17 35 59 89 125

Second differences 12 18 24 30 36

Third differences 6 6 6 6

The third differences, shown in blue in row (3), are all the same constant, 6. Extending row
(3) so that it includes an additional 6 enables us to predict that the next second difference
will be 36. Adding 36 to the first difference 89 gives us the next first difference, 125. Adding
125 to the sixth term 207 yields 332. Using the method of extending the difference table,
we predict that 332 is the next term in the sequence.
Find an nth-Term Formula
Assume the pattern shown by the square tiles in the following figures continues.
a. What is the nth-term formula for the number of tiles in the nth figure of the sequence?
b. How many tiles are in the eighth figure of the sequence?
c. Which figure will consist of exactly 320 tiles?

Solution
a. Examine the figures for patterns. Note the second figure has two tiles on each of the
horizontal sections and one tile between the horizontal sections. The third figure has
three tiles on each horizontal section and two tiles between the horizontal sections.
The fourth figure has four tiles on each horizontal section and three tiles between the
horizontal sections.
 𝑎1 𝑎2 𝑎3 𝑎4
Thus the number of tiles in the nth figure is given by two groups of n plus a group of n less one.
That is,
𝑎𝑛 = 2n + (n-1)
𝑎𝑛 = 3n -1
b. The number of tiles in the eight figure of the sequence is 3(8) -1 = 23.
c. To determine which figure in the sequence will have 320 tiles, we solve the equation
3n -1 = 320
3n = 321 * Add 1 to each side.
n = 107 * Divide each side by 3

The 107th figure is composed of 320 tiles.