 Hi, and welcome to the session. Let us discuss the following question. The question says, if the p of q of and r at terms of a GPR, a, b, and c respectively prove that a to the power q minus r into b to the power r minus p into c to the power p minus q is equal to 1. Let's now begin with the solution. Let a be the first term r be the common ratio of a GP. In the question, it is given that p th term of a GP is equal to a. That means tp is equal to a and tp is equal to a implies a into r to the power p minus 1 is equal to a. Now in the question, it is given that q th term of a GP is equal to b. So that means tq is equal to b and tq is equal to b implies a into r to the power q minus 1 is equal to b and it is also given that r th term of a GP is c. So this means tr is equal to c and this implies a into r to the power r minus 1 is equal to c. We have to prove that a to the power q minus r into b to the power r minus p into c to the power p minus q is equal to 1. Let's name this equation as equation number one. This as 2 and this as 3. Now consider a to the power q minus r into b to the power r minus p into c to the power p minus q. Substitute the value of a, b and c. From 1, we know that a is equal to a into r to the power p minus 1. So a to the power q minus r is equal to a into r to the power p minus 1 whole to the power q minus r. Now substitute the value of b. From 2, we know that b is equal to a into r to the power q minus 1 whole to the power r minus p, c to the power p minus q is equal to a into r to the power r minus 1 whole to the power p minus q. This is equal to a to the power q minus r into r to the power p minus 1 into q minus r. This is equal to a to the power r minus p into r to the power q minus 1 into r minus p and this is equal to a to the power p minus q into r to the power r minus 1 into b minus 2 and this is equal to a to the power q-r into a to the power r-p into a to the power p-q into r to the power p-1 into q-r into r to the power q-1 into r-p into r to the power r-1 into p-q. Since the base is same therefore we can add the powers so this is equal to a to the power q-r plus r-p plus p-q into r to the power on opening the brackets we get p-q minus p-r minus q plus r plus q-r minus p-q minus r plus p plus p-r minus q-r minus p plus q. Now this is equal to a to the power 0 into r to the power 0 and this is equal to 1 into 1 and 1 into 1 is 1. So a to the power q-r into b to the power r-p into c to the power p-q is equal to 1. Hence we have proved that lift inside is equal to write in site. This completes the session. Bye and take care.