Notes On Mathematical Induction - CBSE Class 11 Maths

A statement that has a definite truth value that is, true or false, is called a mathematical statement.

Deductive reasoning involves logical steps to arrive upon a particular case from a general case. Inductive reasoning is the counter-part of deductive reasoning.

This is commonly used in mathematics, where collecting and analysing data is the norm.

This is a process of establishing general principles from particular cases.

In mathematics, we come across many mathematical statements, formulae and theorems that cannot be proved directly.

The principle used to prove mathematical statements, formulae and results involving positive integers, is called the principle of mathematical induction.

To prove mathematical statements through the induction method, certain principles have to be followed. The basic principle behind mathematical induction is that if a statement is true, then the statement that follows the first statement is also true, and so on.

Let P(n) be a statement, where n is a natural number, such that:

The statement is true for n = 1 or P(n) is true.

If the statement is true for n = k, where k is a positive integer, then the statement is also true for n = k + 1.

P(k) is true ⇒ P(k + 1) is true. This is also referred as the inductive step.

Assuming the statement is true for n = k in the inductive step is called inductive hypothesis.

Then, P(n) is true for all natural numbers n.

Principle one is a statement of fact, while principle two is a condition.

If P(n) is true for all n ≥ 2, then step 1 starts from n = 2 and we verify the result of P(2).

If the second principle is true for n = k, then it is also true for n = k + 1.

Ex:

1 + 3 + 5 + ........ + (2n-1) = n2

P(n): 1 + 3 + 5 + ........... + (2n-1) = n2

n = 1,2,3,....

P(1):1 = 12

Thus, P(1) is true.

This statement can be considered as a fact.

Now, take n equal to two.

P(2):1 + 3 = 4 = 22

Thus, P(2) is true.

Next, consider n = 3.

P(3): 1 + 3 + 5

= 9 = 32

Thus, P(3) is true.

Consider P(k) is true.

Need to prove P(k + 1) is true.

1 + 3 + 5 +.........+(2k - 1) ------- (1)


1 + 3 + 5 +.........+ (2k - 1) + [(2k + 1) - 1] = k2 + [(2k + 1) - 1]

= k2 + 2k + 1

=(k + 1)2

=P(k + 1)

P(k + 1) is true.

Hence, P(n) is true for all natural numbers n.

Summary

A statement that has a definite truth value that is, true or false, is called a mathematical statement.

Deductive reasoning involves logical steps to arrive upon a particular case from a general case. Inductive reasoning is the counter-part of deductive reasoning.

This is commonly used in mathematics, where collecting and analysing data is the norm.

This is a process of establishing general principles from particular cases.

In mathematics, we come across many mathematical statements, formulae and theorems that cannot be proved directly.

The principle used to prove mathematical statements, formulae and results involving positive integers, is called the principle of mathematical induction.

To prove mathematical statements through the induction method, certain principles have to be followed. The basic principle behind mathematical induction is that if a statement is true, then the statement that follows the first statement is also true, and so on.

Let P(n) be a statement, where n is a natural number, such that:

The statement is true for n = 1 or P(n) is true.

If the statement is true for n = k, where k is a positive integer, then the statement is also true for n = k + 1.

P(k) is true ⇒ P(k + 1) is true. This is also referred as the inductive step.

Assuming the statement is true for n = k in the inductive step is called inductive hypothesis.

Then, P(n) is true for all natural numbers n.

Principle one is a statement of fact, while principle two is a condition.

If P(n) is true for all n ≥ 2, then step 1 starts from n = 2 and we verify the result of P(2).

If the second principle is true for n = k, then it is also true for n = k + 1.

Ex:

1 + 3 + 5 + ........ + (2n-1) = n2

P(n): 1 + 3 + 5 + ........... + (2n-1) = n2

n = 1,2,3,....

P(1):1 = 12

Thus, P(1) is true.

This statement can be considered as a fact.

Now, take n equal to two.

P(2):1 + 3 = 4 = 22

Thus, P(2) is true.

Next, consider n = 3.

P(3): 1 + 3 + 5

= 9 = 32

Thus, P(3) is true.

Consider P(k) is true.

Need to prove P(k + 1) is true.

1 + 3 + 5 +.........+(2k - 1) ------- (1)


1 + 3 + 5 +.........+ (2k - 1) + [(2k + 1) - 1] = k2 + [(2k + 1) - 1]

= k2 + 2k + 1

=(k + 1)2

=P(k + 1)

P(k + 1) is true.

Hence, P(n) is true for all natural numbers n.

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