What is Mathematical Induction

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Mathematical Induction is basically a technique which is used to establish the proof that a given statement is true for all positive integers or natural numbers. We need to prove two statements which are: First statement in the infinite sequence of statements should be true and if anyone statement in infinite sequence of statements is true then next statement will also be true.

Suppose, we have 'n' natural numbers then P(n) holds all natural numbers so:

0 + 1 + 2 + 3 + 4 + 5 +.........+ n = n (n + 1) / 2,

Where P(n) provides a formula to calculate the sum of natural numbers.

Now we use inductive step to prove the above statement. Where P(k) holds natural number then according to statement P (k + 1) also holds then

(0 + 1+ 2+...................+ k) + (k + 1) = [(k+ 1) ((k+ 1) + 1)] / 2,

After using the induction hypothesis:

k (k + 1)/ 2 + (k + 1),

k (k + 1)/ 2 + (k + 1) = [k(k + 1) + 2(k + 1)] / 2,

= (k² + k + 2k + 2) / 2,

= (k + 1) ( k + 2) / 2,

= [(k + 1) ((k + 1) + 1)] / 2.

So the resulting equation shows that P (k+ 1) also holds.

To understand it, we will take an example: We have an equation 23ⁿ – 1 and we have to prove that it is divisible by 11 for all positive integers 'n'.

So, to prove that we put the value of n = 1 then

23¹ – 1 = 22 which is divisible by 11.

Now, according to induction theorem if P (k) holds then P (k + 1) also holds so

23^{k + 1} – 1= 23. 23^k - 1

= 11. 2. 23^k + (23^k – 1),

Here we can see that resulting equation is divisible by 11.

Multiplying and Dividing Fractions is a concept of algebra.

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