 Hi and welcome to the session, I am Asha and I am going to help you with the following question which says, let S be the sum, P be the product and R be the sum of reciprocals of N terms in a GP, prove that P square into I dash to the power N is equal to S raised to the power N. Let us now begin with the solution and let the GP be A, AR, AR square, AR2 up to N terms. So the sum of these N terms, let us denote it by S will be A into R raised to the power N minus 1 upon R minus 1. Well, let us suppose R is greater than 1. Now let us find the product P of all the terms of GP. So this will be equal to A into AR into AR square into so on up to N terms. So we have A raised to the power N into R raised to the power 1 plus 2 plus up to N minus 1. So this implies A raised to the power N into R raised to the power N into N minus 1 upon 2 since to find the sum of 1 plus 2 plus up to N, this is equal to N into N plus 1 upon 2 and here the last term is N minus 1. So the sum from 1 to N minus 1 is N into N minus 1 upon 2 for the equal to A raised to the power N into R raised to the power N square minus N upon 2. So this is P. Now let the value of S be equation number 1 and the value of P be equation number 2 and now let us find R which is the sum of reciprocal of the term of GP. So 1 upon A plus 1 upon AR plus 1 upon AR square plus 1 up to N terms. Now here the common ratio R is greater than 1 so this implies 1 upon R is less than 1. So R will be equal to 1 upon A into 1 upon R raised to the power N minus 1 upon 1 upon R minus 1 which further implies that 1 upon A into R upon 1 minus R into 1 minus R raised to the power N or this can further be written as 1 minus R raised to the power N upon A into 1 minus R into R raised to the power N minus 1. So this is the value of R and let this be equation number 3. Now we have to show that P square into R raised to the power N is equal to S raised to the power N. So let us start with the left hand side which is P square into R raised to the power N. Now P is raised to the power N into R raised to the power N square minus N upon 2 whole square and R is 1 minus R raised to the power N upon A into 1 minus R into R raised to the power N minus 1 raised to the power N. So this is further equal to A raised to the power 2N into R raised to the power N into N minus 1 into 1 minus R raised to the power N whole raised to the power N upon A raised to the power N 1 minus R raised to the power N into R raised to the power N minus 1 into N further equal to A raised to the power N these two cancels out and gives us A raised to the power N. Now these two also cancels out and we have 1 minus R raised to the power N whole raised to the power N upon 1 minus R raised to the power N which can further be written as A into 1 minus R raised to the power N upon 1 minus R whole raised to the power N A into 1 minus R raised to the power N upon 1 minus R raised to the power S raised to the power N so we have P square into R raised to the power N is equal to S raised to the power N so this completes the session take care and have a good day.