 Hello and welcome to the session. In this session, we will discuss principle of mathematical induction. The word induction basically means generalization from particular cases or facts. In algebra or in other fields of mathematics, there are certain results or statements that are formulated in terms of n, where n is taken a positive integer. To prove such statements, the well-suited principle that is used based on the specific technique is the principle of mathematical induction. According to this, we have that suppose there is a statement pn involving the natural number n such that the first condition is the statement is true for n equal to 1 that is we have that p1 is true. Second condition is if the statement is true for n equal to k where k is some positive integer, then the statement is also true for n equal to k plus 1 that is truth of pk implies the truth of pk plus 1. Then we can say that pn is true for all natural numbers n. The first step in which we prove that p1 is true is called the basic step. And the second step in which we prove that truth of pk implies the truth of pk plus 1. This is the inductive step. Using the principle of mathematical induction, let's try and prove that 1 into 2 plus 2 into 3 plus 3 into 4 plus and so on up to n into n plus 1 is equal to 1 upon 3 multiplied by n multiplied by n plus 1 multiplied by n plus 2. Let this given statement be pn that is we have pn is this statement. First step is the basic step according to which we have to prove that p1 is true. Now for n equal to 1 we have p1 is 1 into 1 plus 1 equal to 1 upon 3 into 1 into 1 plus 1 into 1 plus 2. That is this is the RHS of the given statement and this is the LHS of the given statement. Now from here we get that 2 is equal to 2 which is true hence we have that p1 is true. So we have completed our basic step. Now next we have to do the inductive step according to which we have to show that if pk is true then pk plus 1 is true. So we assume that pk is true for some positive integer k that is we have 1 into 2 plus 2 into 3 plus 3 into 4 plus and so on up to k into k plus 1 is equal to 1 upon 3 multiplied by k multiplied by k plus 1 multiplied by k plus 2. Now we need to prove that pk plus 1 is also true. Now we have 1 into 2 plus 2 into 3 plus 3 into 4 plus and so on up to k into k plus 1 plus k plus 1 into k plus 2 that is we replace k by k plus 1 in the LHS of this statement and so this is equal to 1 into 2 plus 2 into 3 plus 3 into 4 plus and so on up to k into k plus 1 plus k plus 1 into k plus 2. Now using this statement we get this is equal to 1 upon 3 into k into k plus 1 into k plus 2 plus k plus 1 into k plus 2. Now this is equal to k plus 1 into k plus 2 whole multiplied with 1 upon 3 k plus 1. This comes out to be equal to 1 upon 3 into k plus 1 into k plus 2 into k plus 3. That is we get pk plus 1 is the statement 1 into 2 plus 2 into 3 plus 3 into 4 plus and so on up to k plus 1 into k plus 2 is equal to 1 upon 3 into k plus 1 into k plus 2 into k plus 3 and this is true that is we have pk plus 1 is true whenever pk is true. Hence by the principle of mathematical induction we can say that the statement pn is true for all natural numbers n. This is how we use the principle of mathematical induction. This completes the session hope you have understood the principle of mathematical induction.