Free Download of RD Sharma Solutions PDF for Class 11 Maths Chapter 12 For Best Study Support

The technique called mathematical induction is used to prove a theorem, a statement, or a formula that is true for every natural number. Mathematical induction is one of those techniques which can be used to prove a variety of mathematical statements which are formulated in terms of n, where n belongs to the set of positive integers. There are two steps to solve problems of mathematical induction. 

• Base or initial step: It gives proof that the statement is true for an initial value.

• Inductive step: It proves that when a statement is true for an iteration, say nth , then it also holds true for the iteration (n+1)th.

More generally, we can also use the mathematical induction technique to prove that a propositional function, say P(n), is true for all the integers n≥1.

A natural beginning point for proving many mathematical results is to look at a few simple cases like if a statement is true for an initial value. This helps us get some pointers for finding proof. 

We are trying to prove that not one proposition is true, but a whole sequence of propositions are true, one for each n. The trick used in mathematical induction is to prove the given mathematical statement in the sequence, and then prove that if any particular statement is true, then the one after it is also true. This enables us to conclude that all the statements are true in a sequence. 

Solved Examples 

1. Consider if P (n) is a statement “n (n + 1) is even”, then what is P (3)?

Ans:

Given is:

P (n) = n (n + 1) is even.

Therefore

P (3) = 3 (3 + 1)

= 3 (4)

= 12

So, P (3) = 12, P (3) is also even.

2. Consider if P (n) is a statement “n3 + n is divisible by 3”, then prove that P (3) is true, but P (4) is not true.

Ans:

Given is: 

Now, P (n) = n3 + n is divisible by 3

So We have P (n) = n3 + n

Therefore,

P (3) = 33 + 3

= 27 + 3

= 30

P (3) = 30, Hence it is divisible by 3

Now, let’s check with P (4)

P (4) = 43 + 4

= 64 + 4

= 68

P (4) = 68, so it is not divisible by 3

So, P (3) is true and P (4) is not true.

As mentioned earlier, mathematical induction is an easy topic in the Class 11 syllabus when you have the right study material to follow. This is why you need the solution PDF file for all the exercises in this chapter to study and to understand the specific approaches defined by the subject matter experts of Vedantu. Download it today and add it to your study schedule to prepare this chapter better.