 Hi children, my name is Mansi and I am going to help you solve the following question. The question says, prove the following by using the principle of mathematical induction for all n belonging to natural numbers. 1 into 2 into 3, there is 2 into 3 into 4 up till n into n plus 1 into n plus 2 is equal to n into n plus 1 into n plus 2 into n plus 3 the whole divided by 4. In this question we need to prove by using the principle of mathematical induction. Now before doing the solution to this question we see 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 at n such that first P at 1 is true and second statement is true for n equal to k where k is some positive integer, k is true, then statement P at k plus 1 is also true for n equal to k plus 1, then P of 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 start with the solution to this question. Here we have to prove that 1 into 2 into 3 plus 2 into 3 into 4 up till n into n plus 1 into n plus 2 is equal to n into n plus 1 into n plus 2 into n plus 3 the whole divided by 4. Let P of n be 1 into 2 into 3 plus 2 into 3 into 4 and so on till n into n plus 1 into n plus 2 be equal to n into n plus 1 into n plus 2 into n plus 3 the whole divided by 4. Now putting n is equal to 1 P at 1 be 1 into 2 into 3 that is equal to 1 into 1 plus 1 into 1 plus 2 into 1 plus 3 the whole divided by 4 that is equal to 1 into 2 into 3 into 4 the whole divided by 4 that is equal to 1 into 2 into 3 and this is true. Now assuming that P at k is true P at k be 1 into 2 into 3 plus 2 into 3 into 4 and so on till k into k plus 1 into k plus 2 be equal to k into k plus 1 into k plus 2 into k plus 3 the whole divided by 4 and let this be equation number 1. Now to prove that P at k plus 1 is also true we consider P at k plus 1 that is 1 into 2 into 3 plus 2 into 3 into 4 and so on till n into n plus 1 into k plus 2 plus k plus 1 into k plus 2 into k plus 3 be equal to k into k plus 1 into k plus 2 into k plus 3 the whole divided by 4 plus k plus 1 into k plus 2 into k into into k plus 3. And this we get using equation number 1. That is same as k into k plus 1 into k plus 2 into k plus 3 plus 4 into k plus 1 into k plus 2 into k plus 3 the whole divided by 4. That is same as k into k plus 1 into k plus 2 into k plus 3 plus 4 k multiplied by k plus 2 into k plus 3 plus 4 k plus 2 into k plus 3 and this whole divided by 4. This is same as k plus 2 into k plus 3 this whole multiplied by k into k plus 1 plus 4 k plus 4 the whole divided by 4. Now this is same as k plus 2 into k plus 3 multiplied by k square plus 5 k plus 4 and this whole divided by 4. Now k square plus 5 k plus 4 is same as k plus 1 into k plus 4. Thus this becomes equal to k plus 2 into k plus 3 into k plus 1 into k plus 4 the whole divided by 4. This is same as k plus 1 into k plus 1 plus 1 into k plus 1 plus 2 into k plus 1 plus 3 and this whole divided by 4. This becomes 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 hence proved. I hope you understood the question and enjoyed the session. Goodbye.