 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 by 2 plus 1 by 4 plus 1 by 8 up till 1 by 2 raise to the power n is equal to 1 minus 1 divided by 2 raise to the power n. In this question we need to 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. The principle can be explained with the help of two properties if there is a given statement p 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 p at k 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 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 a solution to this question. In this question we have to prove that 1 by 2 plus 1 by 4 plus 1 by 8 up till 1 by 2 raised to power n is equal to 1 minus 1 divided by 2 raised to power n. Now let p at n be 1 by 2 plus 1 by 4 plus 1 by 8 up till 1 by 2 raised to power n is equal to 1 minus 1 by 2 raised to power n. Now putting n equal to 1 p at 1 becomes 1 by 2 is equal to 1 minus 1 by 2 that is same as 1 minus 1 divided by 2 raised to power 1 this is true. Now assuming that p at k is true p at k is 1 by 2 plus 1 by 4 plus 1 by 8 up till 1 divided by 2 raised to power k that is equal to 1 minus 1 divided by 2 raised to power k and this becomes the first equation. Now to prove that p at k plus 1 is also true p at k plus 1 is same as 1 by 2 plus 1 by 4 plus 1 by 8 up till 1 divided by 2 raised to power k plus 1 divided by 2 raised to power k plus 1 that is equal to 1 minus 1 divided by 2 raised to power k plus 1 divided by 2 raised to power k plus 1 and this we get using first equation. Now adding these two expressions we get 1 into 2 raised to power k plus 1 minus 2 plus 1 the whole divided by 2 raised to power k plus 1. Now this is same as 2 raised to power k plus 1 minus 1 the whole divided by 2 raised to power k plus 1. Now this is true p at k plus 1 by 2 divided by 2 raised to power k plus 1 divided by 2 raised to power k plus 1 and 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.