 Hi and welcome to the session. Let us discuss the following question. The question says that with A and 11 terms of a GP are P, Q and S respectively. Show that Q squared is equal to P into S. Now before solving this question, we should know that geometric progression is of the form A, AR, AR squared and so on. The general term of this progression is given by Tn is equal to A into R to the power n minus 1, where A is the first term, R is the common ratio which is obtained by dividing any term by its preceding term. Or we can say that R is obtained by dividing nth term by n minus 1th term. Let's now begin with the solution. Let A be the first term and be the common ratio of a GP. We know that the general term of a geometric progression is given by Tn is equal to A into R to the power n minus 1. Now the fifth term that is T5 is equal to A into R to the power 5 minus 1 and this is equal to A into R to the power 4. And in the question we are given that fifth term is equal to P. So this is equal to P. The eighth term that is T8 is equal to A into R to the power 8 minus 1 and this is equal to A into R to the power 7. And in the question we are given that eighth term is Q. So this is equal to Q. Now the eleventh term that is T11 is equal to A into R to the power 11 minus 1 and this is equal to A into R to the power 10. And in the question we are given that eleventh term is equal to S. So this is equal to S. Let's name this equation as equation number 1. This as 2 and this as 3. Now we have to show that Q squared is equal to P into S. Now consider Q squared. From 2 we know that Q is equal to A into R to the power 7. So Q squared is equal to A into R to the power 7 whole to the power 2. Now this is equal to A to the power 2 into R to the power 14. Now A to the power 2 into R to the power 14 can be written as A into R to the power 4 into A into R to the power 10. From we know that R to the power 4 is equal to P and A into R to the power 10 is equal to S. So using 1 and 3, Q squared is equal to P into S. Hence we have proved that 2 squared is equal to P into S. So this concludes the session. Bye and take care.