 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says The sum of three numbers in GP is 56. If we subtract 1, 7 and 21 from these numbers in that order We obtain an arithmetic progression find the numbers So first let us learn that if we have three numbers a b and c first case is if a b and c are in GP Then b upon a is equal to c upon b or we can say that b square is equal to a c And if these three numbers are in AP then B minus a is equal to c minus b or We can say that 2b is equal to a plus c So with the help of these two ideas We will solve the above problem. So this is our key idea with the solution and let the numbers in GP be a AR and AR square now we are given that some of these three numbers is equal to 56 and also we are given that when 1 7 and 21 are subtracted from a AR and AR square respectively Then the resulting numbers are in that is minus 1 AR minus 7 and AR square minus 21 are in AP and by a key idea we know that if three numbers are in AP then two times of second term is equal to the sum of first and Third term you put two equations Let this be equation number one and this be equation number two Now with the help of these two equations we will find the value of a and our and hence we will find the GP now two can further be written as 2 AR Minus 14 is equal to a plus a our square minus 22 Or we have to air is equal to a plus a our square minus 8 from equation one We have a Plus a our square is equal to 56 Minus a our so on substituting the value of this equation in Hit this equation number three so on substituting this value in equation three Equation three can further be written as to a our is equal to 56 minus a our minus 8 or we have three a our is equal to 48 Which further implies that a our is equal to 16 or a is equal to 16 upon our Now substituting equal to 16 upon our in Equation number one One can be written as 16 upon our Plus 16 upon our into our Plus 16 upon our into our square, which is equal to 56 Or it can be written as 16 Let's 16 our Let's 16 our square is equal to 56 or 16 our square minus 56 are minus 16 hours 40 are and So it is minus 40 our plus 16 is equal to 0 now taking 8 common we have 2 our square minus 5 our Plus 2 is equal to 0 now since 8 is not equal to 0. So this implies 2 our square Minus 5 our plus 2 is equal to 0 Now this by splitting the middle term can be written as 2 our square minus 4 our minus our plus 2 is equal to 0 or Taking 2 our common from the first two terms and Minus one common from the last two terms this can further be written as 2 our Into our minus 2 minus 1 into our minus 2 is equal to 0 or 2 our minus 1 is equal to into our minus 2 Equal to 0 Those two numbers is equal to 0 then at least one of them is 0. So this implies 2 our minus 1 is equal to 0 or R minus 2 is equal to 0 so this implies R is equal to half or R is equal to 2 now if R is equal to half And this implies a Which is 16 upon R will be equal to 16 upon half Equal to 32 and if R is equal to 2 then this implies that a is equal to 16 upon 2 is equal to 8 therefore the GP Is equal to 32 16 and 8 if R is equal to half GP is equal to 8 16 and 32 If R is equal to 2 so the numbers are 8 16 and 32 So this completes the session take care and have a good day