Principle of Mathematical Induction


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If P(n) is a statement (n N); such that:
  • P(1) is true and
  • truth of P(k) implies the truth of P(k+1), then by the principle of

mathematical induction (P.M.I.), the statement P(n) is true for n N.

Working rules for using P.M.I.

Step 1:

Show that the result holds for n = 1.

Step 2:

Assume the validity of the result for n equal to some arbitrary but fixed natural number, say, k.

Step 3:

Show that the result holds for n = k+1.

Step 4:

Conclude that the result holds for all natural numbers.

Using Principle of Mathematical Induction

For any natural number n, prove that

Proof:

We shall use PMI to prove that

Step 1:

Let n = 1

(1) is true for n = 1.

Step 2:

Let (1) be true for n = k

…(2)

Step 3:

Let n = k + 1

LHS of (1)

R.H.S of (1)

\ (1) is true for n = k + 1.

By P.M.I., we have



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