Steps in Mathematical Induction

Steps in Mathematical Induction: It is a concept that helps to prove mathematical results and theorems for all natural numbers. Mathematical thinking is necessary and it helps in understanding problems with this we can reach solutions in a better way. 

The Principle of mathematical induction is a different and specific technique that is used in problems of algebra which can be expressed in terms of n, where n is a natural number. Principle of mathematics connected with deductive reasoning. Deductive reasoning is based on think and fact. That mathematical statement, which is based on the hypothesis that it is true n = 1, and assumed for it is true for n = k then is proved for n = k+1.

Let us understand the concept of the principle of mathematical induction, now we will discuss its statement, its application on various theorems and statements for natural numbers also will discuss the induction step.

Basically, P(n) is the sign of the principle of mathematical induction. It is a technique used to this prove that a mathematical statement holds on all-natural number n = 1,2,3,……………….n. To prove the result of P(n) in that case we use the principle of mathematical induction,

Firstly proved for n = 1, if P(1) true, then we assumed that P(k) is true for some natural number. Now, we use this hypothesis, we will prove for P(k+1) is true. If P(k+1) is true. then the statement will be true for all natural numbers.

Principle of Mathematical Induction Statement:- Now, let us define the Principle of Mathematical Induction and how it use in a statement according to step.

let us suppose that we have a given statement P(n) in which have a natural number such as

• firstly we check statement is true for n = 1 if it is true for n = 1
• then we assumed that it is true for n = k, where k is some natural number,
• then we will prove for n = k+1 if it is true for P(k+1) implies that it is also be true for P(k).

Then, P(n) is true for all natural numbers n.

Now describe through a diagram

Now, before proceeding with the example of the principle of mathematical induction.

let us discuss some important points.

• The first step is a statement of fact. Some mathematical statements hold true for n ≥ 5. In this case, to prove the result using the Principle of Mathematical Induction, step 1 will start from n = 5.
• Step 2 is a conditional property. It does not assert that the statement P(n) is true for n = k. It says that if the statement is true for n = k, then it is also true for n = k+1. In other words, we can say that we assume P(k) is true for some natural number k and then prove that P(k+1) is true.

Mathematical Induction steps

step of this is used to prove the theorem or statement.

Its have 3 main steps of proving the theorem

• Base step:- for this prove for P(n = 1)
• Assumption step:- then assumed that true for P(n=k), k is some natural number.
• The induction step:-Then prove that for P(k+1)

After solving those steps we can say that the ”Principle of mathematical induction is held” for all-natural number n.

Application of Principle of mathematical induction:-

we understood the application of the principle of mathematical induction through

Example:- Prove that the formula for the sum of n natural numbers which is true for all natural numbers, that is, 1 + 2 + 3 + 4 + + …….+ n = n(n+1)/2 using the principle of mathematical induction.

Solution: Let P(n): 1 + 2 + 3 + 4 + 5 + …. + n = n(n+1)/2

Now, we use the concept of mathematical induction and prove this by the induction three steps.

Base Step: To prove P(1) is true.

For n = 1, LHS = 1

RHS = 1(1+1)/2 = 2/2 = 1

Hence LHS = RHS ⇒ P(1) is true.

Assumption Step: Let us assume that P(n) true for n = k, i.e., P(k) is true

⇒ 1 + 2 + 3 + 4 + 5 +….+ k = k(k+1)/2 — (1)

Induction Step: Now we will prove that P(k+1) is true.

To prove: 1 + 2 + 3 + 4 + … + (k+1) = (k+1)(k+2)/2

Let us consider LHS = 1 + 2 + 3 + 4 + … + (k+1)

= 1 + 2 + 3 + 4 + … k + (k+1)

= (1 + 2 + 3 + 4 + … + k) + k+1

= k(k+1)/2 + k+1 [On Using eq. (1)]

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

= (k+1)(k+2)/2 [after taking the common]

= RHS

So, LHS = RHS

⇒ P(n) is true for n = k+1

Hence, by the principle of mathematical induction, P(n) is true for all natural numbers n.

Important Notes on Principle of Mathematical Induction

• In each mathematical we statement is assumed as P(n) for a natural number n.
• Firstly, we will prove for n = 1, then after that, we assume for n = k and then prove for n = k+1.
• The result of the “assumption step” is used after writing the k^th term in the next step (before the (k+1)^th term).
• If we get the RHS from the LHS seems difficult, then do simplify the LHS and the RHS separately and prove that they are equal.