In simple terms Proof of Mathematical Induction is a process typically which is used for proving of the Math statement which is true for all positive integers. Perhaps we must not get confused with the proof by mathematical induction with the non-rigorous reasoning of mathematics, whereas it is a rigorous deductive reasoning of mathematics. We can understand this by taking an example;
0 + 1 + 2 + 3 + 4 + ····· + n = n (n + 1)/2. (This is true for all natural number).
Now we have to prove that this statement holds true for all the positive integers.
Put n = 1,
1 = 1(1 + 1) / 2,
As we can observe that on the left hand side of this equation there exists only single term that is 1 while on the right hand side if we are going to solve that part then the outcome will be 1. Now it is clear that both sides i.e. the left hand side (LHS) and the right hand side (RHS) are equal. Therefore this statement holds true to all n Є 1 (natural number belongs to 1) then it is clear that this statement will also be true for every n Є n+1.
Here two things should be noted that while conducting the mathematical induction proofs the proving takes place with two important steps, first one is the basic step which says that the statement must be true for all positive integers and then the other step says that which is a inductive step is, this statement will also hold true for all positive integers which belong to n + 1. This method of proving in math induction is helpful for proving the statement true and after the completion of first step the next step is automatically concluded, this may remind you of domino effect (this effect is a chain sequence effect which takes place with a small simple change), if the first one fails the other simply by itself gets discharged.
Number system Numbers can be divided into two categories i.e. even numbers and Odd Numbers. Even numbers are numbers which when divided by 2 give zero as remainder and when we divide odd numbers by two we get remainder as 1. Now let us see what sum of odd numbers is:
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Counting is a crucial part of mathematics. There are several methods like Permutation, combination and Factorization which we follow to count the number of possibilities. Counting regions is also one such kind of problem we may find in Math.
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