 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 3 plus 2 into 3 square plus 3 into 3 cube up till n into 3 raised to the power n is equal to 2n minus 1 multiplied by 3 raised to the power n plus 1 plus 3 the whole divided by 4. In this question we prove by using the principle of mathematical induction. Now before doing the 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. This can be stated with the following 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 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 start with the solution to this question. Here we have to prove that 1 into 3 plus 2 into 3 square plus 3 into 3 cube so on till n into 3 raise to power n is equal to 2n minus 1 multiplied by 3 raise to power n plus 1 plus 3 the whole divided by 4. Let p at n be 1 into 3 plus 2 into 3 square plus 3 into 3 cube up till n into 3 raise to power n that is equal to 2n minus 1 into 3 raise to power n plus 1 plus 3 the whole divided by 4. Now putting n equal to 1 p at 1 becomes 1 into 3 that is equal to 2 minus 1 into 3 raise to power 1 plus 1 plus 3 the whole divided by 4 that is equal to 12 divided by 4 and that is equal to 3 and this is true. Now assuming that p at k is true p at k becomes 1 into 3 plus 2 into 3 square plus 3 into 3 cube and so on till k into 3 raise to power k that is equal to 2k minus 1 into 3 raise to power k plus 1 plus 3 and the whole divided by 4. Now to prove that p at k plus 1 is also true. Now p at k plus 1 is 1 into 3 plus 2 into 3 square plus 3 into 3 cube up till k into 3 raise to power k plus k plus 1 into 3 raise to power k plus 1 this is same as 2k minus 1 the whole multiplied by 3 raise to power k plus 1 plus 3 the whole divided by 4 plus k plus 1 into 3 raise to power k plus 1. This is same as 2k minus 1 into 3 raise to power k plus 1 plus 3 plus 4 into k plus 1 into 3 raise to power k plus 1 and the whole divided by 4. This is same as 3 raise to power k plus 1 into 2k minus 1 plus 4k plus 4 plus 3 and the whole divided by 4. This is same as 3 raise to power k plus 1 into 6k plus 3 plus 3 the whole divided by 4. Now this is same as 3 raise to power k plus 1 multiplied by 3 into 2k plus 1 plus 3 the whole divided by 4. This is same as 3 raise to power k plus 1 plus 1 into 2k plus 1 minus 1 plus 3 the whole divided by 4 and we see that this is same as p at k plus 1 hence from 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.