 Hi and welcome to the session. Let us discuss the following question. The question says if the first and end term of a GP are A and B respectively and if P is the product of N terms proves that P squared is equal to AB to the power N. Let's now begin with the solution. Let R be the common ratio of a GP. The question it is given that first term of a GP is A that means T1 is equal to A and nth term of a GP is equal to P that means TN is equal to B. Now TN is equal to B implies A into R to the power N minus 1 is equal to B. Now it is also given in the question that P is the product of N terms. So this means P is equal to T1 into T2 into T3 and so on up to TN. Since A is the first term and R is the common ratio therefore T1 that is first term is equal to A, T2 that is second term is equal to AR, third term that is T3 is equal to AR squared and so on TN is equal to A into R to the power N minus 1. Now this is equal to A to the power N since A appears N times into R to the power 1 plus 2 plus so on up to N minus 1. We know that sum of N natural numbers that is 1 plus 2 plus 3 up to so on up to N terms is equal to N into N plus 1 by 2 therefore sum of N minus 1 natural numbers that is 1 plus 2 plus 3 plus so on up to N minus 1 is equal to N minus 1 into N minus 1 plus 1 by 2 and this is equal to N minus 1 into N by 2. So now this expression can be written as A to the power N into R to the power N minus 1 into N by 2. So P is equal to A to the power N into R to the power N minus 1 into N by 2. We have to prove that P squared is equal to AB to the power N so let's now consider P squared. Now P squared is equal to A to the power N into R to the power N minus 1 into N by 2 whole square and this is equal to A to the power 2N into R to the power N minus 1 into N and this is equal to A to the power 2 into R to the power N minus 1 whole to the power N and this can be written as A into A into R to the power N minus 1 whole to the power N. Now we know that A into R to the power N minus 1 is equal to B. So we will now substitute B in place of A into R to the power N minus 1 in this expression. So by substituting B we get A into B to the power N. So P squared is equal to AB to the power N. Hence we have proved that P squared is equal to AB to the power N. This completes this session. Bye and take care.