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The Principle of Mathematical Induction

There are many ways to prove theorems in mathematics. One of such is mathematical induction which can be used easily to prove many of other theorems. The method is quite easier from other methods of proving a theorem or statement in mathematics. On similar grounds we can also prove the statements of sum of squares of first n natural numbers to be n (n + 1) (2n + 1) / 6, the sum of first n odd numbers to be n^2 or the sum of cubes of first n natural numbers to be [n (n + 1) / 2]^2 and many more. The simple definition for the rule of the mathematical induction is: “Suppose a mathematical statement is true for the first value of applied to the statement. Assuming that it is also true for ‘kth’ value as well, if we prove that it is valid for (k + 1)th value, then by the principle of mathematical induction the statement holds true for all natural numbers ‘n’.” The principle of mathematical induction is mostly used in cases of natural numbers only.

Let us understand mathematical induction proof with the help of an example, we know that the sum of ‘n’ natural numbers is [n (n + 1) / 2. 
SOLUTION:
We need to prove that 1 + 2 + 3 + … + n = [n (n + 1)] / 2.
Split proof in 3 steps to make solution more simple and understandable
Step 1: For n = 1
LHS = 1
RHS = 1 (1 + 1) / 2 = 2 / 2 = 1 = LHS
Result is true for n = 1

Step 2: Assumption
Assume that, the statement is true for n = t
1 + 2 + 3 + … + t = t (t + 1) / 2

Step 3: Prove statement is true for n = t + 1 using fact of step 2
1 + 2 + 3 + … + t + (t + 1) = (t + 1) (t + 2) / 2.

LHS = 1 + 2 + 3 + … + t + (t + 1)
 = [t (t + 1) / 2] + (t + 1)
 = [t (t + 1) + 2 (t + 1)] / 2
 = (t + 1) (t + 2) / 2
 = RHS
The statement is true for n = t + 1.

Hence, by the principle of mathematical induction result is proved.

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