 Hi children, my name is Mansi and I'm 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, 2n plus 7 is strictly less than n plus 3 the whole square. In this question you have to prove by using the principle of mathematical induction. Now before starting the solution we see the key idea behind the question. Now 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 if statement is true for n equal to k where k is some positive integer p at k is true then statement p at k plus 1 is also true for n equal to k plus 1 then e at n is true for all natural numbers. 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 and 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 proof to this question here we have to prove that 2n plus 7 is strictly less than n plus 3 the whole square let p at n be 2n plus 7 is strictly less than n plus 3 the whole square quoting n equal to 1 p at 1 becomes 2 into 1 plus 7 is strictly less than 1 plus 3 the whole square or 9 is strictly less than 16 and this is true assuming p at k is true p at k is same as 2k plus 7 is strictly less than k plus 3 the whole square and this becomes our first equation now to prove that p at k plus 1 is also true putting n equal to k plus 1 we have to prove that p at k plus 1 is twice of k plus 1 plus 7 is strictly less than k plus 1 plus 3 the whole square that is equal to 2k plus 9 is strictly less than k plus 4 the whole square now starting with 1 k plus 7 is strictly less than k plus 3 the whole square now adding 2 to both sides we get k plus 7 plus 2 is strictly less than k plus 3 the whole square plus 2 this further implies that 2k plus 9 strictly less than k square plus 6k plus 9 plus 2 now expressing in terms of k plus 4 the whole square we get 2k plus 9 is strictly less than k square plus 8k plus 16 minus 2k plus 5 this implies 2k plus 9 is strictly less than k plus 4 the whole square minus 2k plus 5 now since 2k plus 9 is less than k plus 4 the whole square minus a positive number this implies that 2k plus 9 is also less than k plus 4 the whole square thus k plus 9 is strictly less than k plus 4 the whole square 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 hence proved I hope you understood the question and enjoyed the session goodbye.