 Hi children, my name is Mansi and I'm going to help you solve the following question. The question says, approve the following by using the principle of mathematical induction for all n belonging to natural numbers. 1 into 2 plus 2 into 3 up till n into n plus 1 is equal to n into n plus 1 into n plus 2 the whole divided by 3. In this question we prove by using the principle of mathematical induction. Now before starting the solution we see first the key idea behind the question. We know that the principle of mathematical induction is a specific technique which is used to prove certain statements that are formulated in terms of n where n is a positive integer. The principle can be explained with the help of two properties. If there is a given statement P n such that first P at 1 is true and second if statement is true for n equal to k where k is some positive integer. So the principle k is true then statement P at k plus 1 is also true for n equal to k plus 1. Then P at n is true for all natural numbers n. Using these two properties we will show that statement is true for n equal to 1. Then assume it is true for n equal to k. Then we prove it is also true for n equal to k plus 1. Hence proving that it is true for all n belonging to natural numbers. Now we see the solution to this question. Here we have to prove that 1 into 2 plus 2 into 3 up till n into n plus 1 is equal to n into n plus 1 into n plus 2 the whole divided by 3. Let P of n be 1 into 2 plus 2 into 3 so on till n into n plus 1 is equal to n into n plus 1 into n plus 2 the whole divided by 3. Now putting n equal to 1 P at 1 becomes 1 into 2 that is equal to 1 into 1 plus 1 into 1 plus 2 the whole divided by 3 that is same as 1 into 2 into 3 the whole divided by 3 and this is same as 1 into 2 that is equal to 2 and this is true. Now assuming that P at k is true P at k becomes 1 into 2 plus 2 into 3 plus 3 into 4 and so on till k into k plus 1 be equal to k into k plus 1 into k plus 2 and the whole divided by 3. Now to prove that P at k plus 1 is also true P at k plus 1 is 1 into 2 plus 2 into 3 plus 3 into 4 and so on till k into k plus 1 plus k plus 1 into k plus 2. This is equal to k into k plus 1 into k plus 2 the whole divided by 3 plus k plus 1 into k plus 2. This is equal to k into k plus 1 into k plus 2 the whole divided by 3 plus k plus 1 into k plus 2. This is same as k into k plus 1 into k plus 1 into k plus 2 plus 3 into k plus 1 into k plus 2 and this entire term divided by 3. Now taking common multiple k plus 1 into k plus 2 this becomes equal to k plus 1 into k plus 1 into k plus 2 into k plus 3 the whole divided by 3. Now this is written as k plus 1 into k plus 1 plus 1 into k plus 1 plus 2 the whole divided by 3. This is same as P at k plus 1 thus P at k plus 1 is true wherever P at k is true. Hence from the principle of mathematical induction the statement P at n is true for all natural numbers n hence proved. So I am going to I hope you understood the question and enjoyed the session. Goodbye.