Basic Mathematics for College Students

A-23

Inductive and Deductive

Reasoning

APPENDIX

III

SECTION III.1

Inductive and Deductive Reasoning

Objectives

1 Use inductive reasoning to

solve problems.

2 Use deductive reasoning to

solve problems.

To reason means to think logically. The objective of this appendix is to develop your

problem-solving ability by improving your reasoning skills. We will introduce two

fundamental types of reasoning that can be applied in a wide variety of settings. They

are known as inductive reasoningand deductive reasoning.

Use inductive reasoning to solve problems.

In a laboratory, scientists conduct experiments and observe outcomes. After several

repetitions with similar outcomes, the scientist will generalize the results into a

statement that appears to be true:

• If I heat water to 212°F, it will boil.


• If I drop a weight, it will fall.


• If I combine an acid with a base, a chemical reaction occurs.
  When we draw general conclusions from specific observations, we are using
  inductive reasoning.The next examples show how inductive reasoning can be used in
  mathematical thinking. Given a list of numbers or symbols, called a sequence,we can
  often find a missing term of the sequence by looking for patterns and applying
  inductive reasoning.

1

Self Check 1

Find the next number in the
sequence 3,1,1,3,....

Now TryProblem 11

EXAMPLE (^1) Find the next number in the sequence 5, 8, 11, 14,....
StrategyWe will find the differencebetween pairs of numbers in the sequence.
WHYThis process will help us discover a pattern that we can use to find the next
number in the sequence.
Solution
The numbers in the sequence 5, 8, 11, 14,...are increasing. We can find the
difference between each pair of successive numbers as follows:
8  5  3 Subtract the first number, 5, from the second number, 8.
11  8  3 Subtract the second number, 8, from the third number, 11.
14  11  3 Subtract the third number, 11 from the fourth number, 14.
The difference between each pair of numbers is 3. This means that each number in
the sequence is 3 greater than the previous one. Thus, the next number in the
sequence is 14 3, or 17.
EXAMPLE (^2) Find the next number in the sequence 2,4,6,8,....
StrategyThe terms of the sequence are decreasing. We will determine how each
number differs from the previous number.
WHYThis type of examination helps us discover a pattern that we can use to find
the next number in the sequence.