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Principle of Mathematical Induction

As mathematical approach to prove so many things, the mathematical induction plays a huge role and in the most possible way it’s important for proving the natural numbers’ property (n).

As per mathematical induction principle, X(n) property is same for all the natural numbers – 0,1,2,3, ………n.

Now, considering the given statement X (n), it involves n natural numbers in the following cases:

  1. The statement holds true and valid for the value n = 1; X(1) is true,
  2. If statement for n = k is true, where k exist as a positive integer value, then this statement even holds true for all n= k+1 cases that is, X(k) resulting in X(k+1) truth.

From this, it’s understood that for all the ‘n’ natural numbers, X(n) exist true.

Property a: Simply, from what’s mentioned above is the statement based on fact. In the conditions, where statement holds true for cases including n≥5, accordingly step ‘a’ will going to range from n = 5 & we then thus prove the result that is defined for n = 5, that is, X(5).

Property b: It’s the conditional property because it doesn’t prove the statement stands true for the value of n = k, although if it’s true in the context of n = k, then it’s going to be dominate as true for the value of n = k+1.

Thus, to verify a property, we first need to prove or verify the conditional proposition/suggestion/hypothesis. To prove the conditional proposition, it’s known as theorem’s inductive step.

The assumption in which the given statement on the condition, n = k is true is defined as an inductive hypothesis.

Example

In algebra, it does follow a particular pattern, a set of numerous patterns is as follows:

1 = 1² = 1

9 = 3² = 1 + 3 + 5

25 = 5² = 1 + 3 + 5 + 7 + 9

36 = 6² = 1 + 3 + 5 + 7 + 9 + 11

From the example of the above-mentioned pattern, here’s the conclusion that – 2nd natural number’s square is equal to first 2 odd numbers’ sum.

Similarly, first 3 odd n natural numbers’ sum exist equal to 3rd natural numbers’ square.

From the pattern shown in the above example, the conclusion says:

“1 + 3 + 5 + 7 + ……+ (2n – 1) = n²” , or square of n² is defined to be equal for the first odd natural numbers’ sum, which can be written as:

X(n) = 1 + 3 + 5 + 7 + 9 +…….+ (2n-1) = n²

Taking mathematical induction principle’s help, it’s required to verify concerning X(n) holds true for all n values.

Talking of this process, this means that X(1) value is true. This step I is known as the basic or base step. As this step establish the mathematical induction’s base. 1 = 1², X(1), thus is true.

Next step to follow is base which is known as the “Inductive Step”. Further in this step, we have supposed that X(m) value is real for any of the positive integer “m”, it’s needed for us to prove – X(m+1) is also true. Because X(m) holds true as:

1 + 3 + 5 + 7…. + (2m-1) = m² Eq (1)

Now considering, 1 + 3 + 5 + 7 +…..+ (2m-1) + {2(m+1)-1} Eq (2)

By using (1), we get

= m² + (2m + 1) = (m + 1)²

Hence, even X(m + 1) is true and therefore, resulting in the proof of the inductive step. This gives that conclusion that for all natural numbers, X(n) is true.

So, from the above example, it’s been proven that for all n natural numbers, X(n) holds true.

Applications

Deduction

Deduction refers to the statement which needs to get proved plus is usually known as the mathematical theorem. Using deductive steps, you can prove the theorems or disprove them; also known as the conjectures.

The deduction is known to be the application considering a general case up to a specific case.

Inductive Reasoning

The opposite of deductive reasoning is called Inductive Reasoning. It’s dependent on particular case study and to observe those incidents which leads to that particular case.

Reasoning by induction forms scientific reasoning basis and if it’s generally in the use of mathematics. Considering the collection, information on analyzing is a measure in mathematical induction, scientific reasoning. The mathematical induction also make use of the same standards and measures for generalizing the particular cases or facts.

Also, mathematical induction principle has its use in algebra or other mathematical streams which includes statements regarding ‘n’ or result formulation.

 

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