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Mathematical Induction

There are many theorems and statements in mathematics that are to be proved with the help of mathematical induction. This is a way of proving statements that those statements are true for all positive integers.

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Strategy of Mathematical Induction

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The technique of mathematical induction is based on the principle that if a statement is true for integers - 1, 2, 3 and so on, up to k, then it is also true for all integers.

Following steps are to be followed while proving a statement by mathematical induction -

Let us consider that we are to prove that a statement P(n) holds for all positive integers.

Step 1:
Prove that P(1) holds, i.e. P(n) holds for n = 1.

Step 2:
Now, assume that the statement P does hold for some arbitrary integer (say k). i.e. P(k) holds.

Step 3:
Assuming P(k) is true, prove that the statement P holds for a successor of k, i.e. prove that the P (k + 1) holds.

Step 4:
If P (k + 1) holds, then it can be concluded that the statement holds for all integers. This means that P(n) is true.

Examples: 

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1) Using mathematical induction, prove that sum of n positive integers is $\frac{n}{2}$$(n+1)$.

Solution:
We have to prove that 1 + 2 + 3 + 4 + ...... + n =
$\frac{n}{2}$$(n+1)$

To prove that above statement is true for n = 1

First integer = 1 whose sum will also be 1.

Therefore, it holds for n = 1

Now, we can assume that given statement is true for k positive integers

Hence,
1 + 2 + 3 + 4 + ...... + k = $\frac{k}{2}$$(k+1)$

We shall prove the validity of statement for k + 1

Sum of $k\ +\ 1$ positive integers

= 1 + 2 + 3 + 4 + .... + (k + 1)

= (1 + 2 + 3 + 4 + .... + k) + (k + 1)

We know the sum of k positive integers

=
$\frac{k}{2}$$(k\ +\ 1)\ +\ (k\ +\ 1)$

= $\frac{k\ (k\ +\ 1)\ +\ 2(k\ +\ 1)}{2}$

=
$\frac{(k\ +\ 1)(k\ +\ 2)}{2}$

= $(k+1)$ $\frac{[(k\ +\ 1)\ +\ 1]}{2}$

This proves that statement holds for $n$ = $k\ +\ 1$.

Therefore, we conclude that the statement holds of all n positive integers.

More topics in  Mathematical Induction Formula
Principle of Mathematical Induction
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