 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 41 raise to power n minus 14 raise to power n is a multiple of 27. In this question we need to prove by using the principle of mathematical induction. Now before starting the solution 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 if statement is true for n equal to k where k is some positive integer pk 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 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 solution to this question here we have to prove that 41 raise to power n minus 14 raise to power n is a multiple of 27. Now let p at n be 41 raise to power n minus 14 raise to power n is a multiple of 27. Now putting n equal to 1 p at 1 becomes 41 raise to power 1 minus 14 raise to power 1 that is equal to 27 and we see that 27 is a multiple of 27 thus p at 1 is true. Now assuming that p at k is true p at k becomes 41 raise to power k minus 14 raise to power k is a multiple of 27. Now we write 41 raise to power k minus 14 raise to power k is equal to 27 d where d belongs to n and this becomes the first equation. Now to prove that p at k plus 1 is also true putting n equal to k plus 1 we find 41 raise to power k plus 1 minus 14 raise to power k plus 1. Now first grouping the numbers according to common multiples we get 41 raise to power k plus 1 minus 41 raise to power k into 14 plus 41 raise to power k into 14 minus 14 raise to power k plus 1. Now taking the common multiples out of the brackets we get 41 raise to power k into 41 minus 14 plus 14 multiplied by 41 raise to power k minus 14 raise to power k this is equal to 27 into 41 raise to power k plus 14 into 27 d and this we get using first. Now this is same as 27 multiplied by 41 raise to power k plus 14 d now we see that the above expression that is 27 multiplied by 41 raise to power k plus 14 d is a multiple of 27 thus 41 raise to power k plus 1 minus 14 raise to power k plus 1 is equal to p at k plus 1 and this is true hence from the principle of mathematical induction the statement p at n is true for all natural numbers n hence proved. I hope you understood the question and enjoyed the session goodbye.