mathematical induction principle

Let us learn the principle of mathematical induction,
THE NATURAL NUMBERS are the counting numbers: 1, 2, 3, 4, etc. Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number.

By "every" natural number, or "all" natural numbers, we mean any one that we might possibly name.

For example,

1 + 2 + 3 + . . . + n = ½n(n + 1).

This asserts that the sum of consecutive numbers from 1 to n is given by the formula on the right. We want to prove that this will be true for n = 1, n = 2, n = 3, and so on. Now we can test the formula for any given number, say n = 3:

1 + 2 + 3 = ½· 3· 4 = 6

-- which is true. It is also true for n = 4:

1 + 2 + 3 + 4 = ½· 4· 5 = 10

But how are we to prove this rule for every value of n?

The method of proof is the following. It is called the principle of mathematical induction.

If

1) when a statement is true for a natural number n = k,
then it will also be true for its successor, n = k + 1;

and

2) the statement is true for n = 1;

then the statement will be true for every natural number n.

For, when the statement is true for n = 1, then according to 1), it will also be true for 2. But that implies it will be true for 3; which implies it will be true for 4. And so on. It will be true for any natural number that we might name.

To prove a statement by induction, then, we must prove parts 1) and 2) above.

The hypothesis of Step 1) -- "The statement is true for n = k" -- is called the induction assumption, or the induction hypothesis. It is what we assume when we prove a theorem by induction.
Hope the above explanation helped you.