 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question is, has proved the following by using the principle of mathematical induction for all n belonging to natural numbers. 1 plus 2 plus 3 up till n is strictly less than 1 by 8 into 2 n plus 1 the whole square. So, let us start with the solution to this question. Here we have to prove that 1 plus 2 plus 3 up till n is strictly less than 1 by 8 into 2 n plus 1 the whole square. Let p at n be 1 plus 2 plus 3 and so on till n is strictly less than 1 by 8 into 2 n plus 1 the whole square. Now putting n equal to 1, p at 1 becomes 1 is strictly less than 1 upon 8 into 2 into 1 plus 1 the whole square or 1 is strictly less than 9 upon 8 or 1 is strictly less than 1 1 by 8 and we see that this is true. Now assuming that p at k is true, p at k becomes 1 plus 2 plus 3 and so on till k is strictly less than 1 upon 8 into 2 k plus 1 the whole square and this is the first equation. Now to prove that p at k plus 1 is also true putting n equal to k plus 1 we have to find 1 plus 2 plus 3 and so on till k plus k plus 1. Now putting n equal to k plus 1 implies 1 plus 2 plus 3 till k plus k plus 1 is strictly less than 1 upon 8 into 2 2 k plus 1 the whole square plus k plus 1 and this we get using first. Now adding the two expressions we get 1 plus 2 plus 3 and so on till k plus k plus 1 is strictly less than 4 k square plus 4 k plus 1 plus 8 into k plus 1 the whole divided by 8. This also implies that 1 plus 2 plus 3 till k plus k plus 1 is strictly less than 4 k square plus 12 k plus 9 the whole divided by 8. Now we see that 4 k square plus 12 k plus 9 is equal to 2 k plus 3 the whole square so we get 1 plus 2 plus 3 till k plus k plus 1 is strictly less than 2 k plus 3 the whole square divided by 8. This further implies that 1 plus 2 plus 3 up till k plus k plus 1 is strictly less than 1 by 8 into 2 into k plus 1 plus 1 the whole square that is equal to 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. So I hope you understood the question and enjoyed the session have a good day.