Principle of Mathematical Induction
Q What do you mean by principle?
Ans According to google,
a basic general law, rule or idea is known as principle.
Q What is induction?
a means of proving a theorem by showing that if it is true of any particular case it is true of the next case in a series, and then showing that it is indeed true in one particular case.
Humein pata hai ki electricity induce ho sakti hai, iska matlab hai ek device se doosre tak electricity ko transfer karna, similarly yahan par bhi induction kuch aisi hi hai.
Let’s start our journey by observing a pattern. Do you know the basic step of finding a formula or theorem is observing the pattern!!
Let’s start with observing the squares of first five numbers.
Now let us suppose the following series:
Q Can we guess the next no.?
Ans let’s see,
The next no. must be 25+11
That is 36
And of course it is true as square of six is 36.
Hence we noticed that
the sum of first two odd no. is the square of second natural no. that is 2.
Similarly, we can write 4+5 as 1+3+5
So, sum of first three odd no.s is the square of third natural no. that is 3.At the end, we may conclude that
the sum of first n odd no.s is the square of n th natural no.
Now we have defined a general rule.
But the question arises here is that if it is true for all natural no.s?
The only answer to this question is
to prove it.
We can’t check each and every natural no. which will may end to infinity!!!
So, if we really want to prove it, we can just prove that if it is true for n=k where k is any natural no., then it is also true for n=k+1, means it’s next term.
Now if we prove the above statement only, our rule becomes true for any natural no.
By proving the above statement, we have just proved that it is true for n=K+1 and also we have assumed that it is true for n=k, so if we talk about n=k+2, means then next term, then k+2 becomes k+1+1 and let k+1=t, so as it is true for t then it is also true for t+1 by the above proof. Hence, we can move on and on and this rule will become true for any natural no.
Now let’s prove our rule by this method.
P(n): 1+3+ 5+7+…..till (2n-1) =n^2
Why the last term is (2n-1)?
We know that first term is 1
The second term is 3 which is 4-1
The third term is 5 which is 6-1
So, the nth term is 2*n-1
Now the basic step is to prove that it is true for n=1
Means the sum of first odd no. is square of first natural no., which is obvious as 1=1^2
Now let us suppose that the given rule is true for n=k.
P(k): 1+3+5+7+…..+(2*k-1) = k^2
Now we have to prove that
Which can be written as
Because k^2 is the sum till the previous term
Now LHS can be written as k^2+2*k+1 which is equal to RHS.
Hence,our rule is true for any natural no.
Did you know that this method of proving a rule or theorem is known as principle of mathematical induction!!
Now we can state this principle according to ncert as
Suppose there is a given statement P(n) involving the natural number n such that
(i) The statement is true for n = 1, i.e., P(1) is true, and
(ii) If the statement is true for n = k (where k is some positive integer), then
the statement is also true for n = k + 1, i.e., truth of P(k) implies the
truth of P (k + 1).
Then, P(n) is true for all natural numbers n.