 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 plus 3 plus 3 squared up to 3 raised to the power n minus 1 is equal to 3 raised to the power n minus 1 divided by 2. In this question we prove by using the principle of mathematical induction. Now before doing 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 of n such that first p of 1 is true and second if statement is true for n equal to k where k is some positive integer of k is true. Then statement p at k plus 1 is also true for n is 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 is equal to 1 then assume it is true for n is equal to k then we prove it is also true for n is 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 we have to prove that 1 plus 3 plus 3 square up till 3 raised to power n minus 1 is equal to 3 raised to power n minus 1 the whole divided by 2. Let p of n be 1 plus 3 plus 3 square up till 3 raised to power n minus 1 is equal to 3 raised to power n minus 1 the whole divided by 2 putting n equal to 1 p at 1 becomes equal to 1 which is also equal to 3 raised to power 1 minus 1 divided by 2 which is equal to 2 by 2 that is 1 and this is true. Now assuming that p of k is true p of k is 1 plus 3 plus 3 square up till 3 raised to power k minus 1 that is equal to 3 raised to power k minus 1 the whole divided by 2. Now to prove that p at k plus 1 is also true we consider p at k plus 1 which is equal to 1 plus 3 plus 3 square up till 3 at k minus 1 plus 3 at k which is equal to 3 raised to power k minus 1 the whole divided by 2 plus 3 raised to power k which is again equal to 3 raised to power k minus 1 plus 2 multiplied by 3 raised to power k and the whole divided by 2. It is equal to 3 raised to power k multiplied by 1 plus 2 to minus 1 the whole divided by 2 and this is equal to 3 raised to power k multiplied by 3 minus 1 the whole divided by 2 which is again equal to 3 raised to power k plus 1 minus 1 divided by 2. Thus p at k plus 1 is true wherever p at k is true. Hence from the principle of mathematical induction the statement at n is true for all natural numbers n. Hence the result. I hope you understood the question and enjoyed the session. Goodbye.