Question

Explain the principle of mathematical induction with the help of example

Answer 1

In base step the statement is to be proved for an initial value of natural numbers. Normally 0 or 1 is used to prove the statement. In inductive step first thing is to assume that the statement is true for kth iteration, using this assumption second thing is to prove that it is true for (k+1)th iteration. So after the statement is established for above two steps, then it can be induced it is true for any iteration or all the numbers in the statement. The assumption in the inductive step is termed as Induction Hypothesis.

1 + 2 + 3 + 4 + ..… + n = {n(n + 1)}/2.

For all positive integers n.

We denote this mathematical statement by P(n). If n = 1, then we find that {1(1 + 1)}/2= 1.

Hence the above statement is true for n = 1.

Suppose n = 2. Then 1 + 2 = 3 and {2(2 + 1)}/2= 3. So we find that the statement is true for n = 2.

Thus, it is easy to verify that P(n) is true for n = 1, 2, 3 or 4. But it is impossible to verify that the above statement is true for all positive integers