Principle of Mathematical Induction, Objectives, What is a Statement? the Principle of Mathematical Induction

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Objectives

After studying this lesson, you will be able to:

After studying this lesson

After Studying this Lesson

What is a Statement?

In your daily interactions, you must have made several assertions in the form of sentences.

Of these assertions, the ones that are either true or false are called statement or propositions.

For instance,

“I am years old” and “If , then ” are statements, but ‘When will you leave ?’ And ‘How wonderful!’ are not statements.

  • Notice that a statement has to be a definite assertion which can be true or false, but not both.

For example, is not a statement, because we don’t know what , is. If , it is true, but if , ‘it is not true. Therefore, is not accepted by mathematicians as a statement.

But both and for any real number ’ are statements, the first one true and the second one false.

Example:

If denotes , write and , where .

Solution:

Replacing by and , respectively in , we get

i.e.

i.e.

The Principle of Mathematical Induction

Let be a statement involving a natural number . If

  1. It is true for , i.e., ) is true; and

  2. Assuming and to be true, it can be proved that is true; then must be true for every natural number .

Note:

That condition (2) above does not say that is true. It says that whenever is true, then is true.

Example:

Using principle of mathematical induction, prove that is a natural number for

all natural numbers .

Solution:

Let be a natural number.

is a natural number.

Or , which is a natural number

is true.

Let is a natural number be true

Now,

By (1) is a natural number.

Also is a natural number and is also a natural number.

being sum of natural numbers is a natural number.

is true, whenever is true.

is true for all natural numbers .

Hence, is a natural number for all natural numbers .

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