 p s q s r s terms of g p r x y z find x to power q minus r y to power r minus e z to power p minus q x is equal to a let's capital let's say capital r is the ratio because small r is given a r and this is p minus 1 is it x is a r p minus 1 so this is x what is y now here itself do the first thing x is equal to this much so x to the power q minus r this part okay finish this here itself so you'll get x to the power q minus r is equal to a r p minus 1 whole q minus r which is nothing but a to the power q minus r and r p minus 1 q minus r right this is for x to the power q minus r similarly y to power r minus p will be a r minus p times r what was this q s term so q minus 1 r minus p right and then similarly z p minus q is a a p minus q into capital r r minus 1 and p minus right now you have to multiply these three LHS and if you multiply what will you get you will get a q minus r times r and expand the brackets simultaneously p q minus p r minus q plus r right into a r minus p into r q r minus q p minus r plus p into a p minus q into r r p minus r q minus p plus q right so all powers of a will get added because it's a multiplication so q it is a q minus r plus r minus p plus p minus r like that so r sorry what is that p minus q p minus q so p and minus p q and minus q r and minus r all so this is a to the power 0 similarly if you see powers of r it will also come out to be 0 so a to the power 0 into r to power 0 1 again question is if in a GP of positive terms if any term is equal to the sum of the next two terms find r right so if in a GP of positive terms if any of the term is equal to the sum of the next two terms that means a n is equal to a n plus 1 plus a n plus 2 this is the so this is right a n is equal to a n plus 1 plus a n plus 2 so what is a n a r to the power n minus 1 is equal to a r to the power n plus a r n plus 1 right so a will get all over positive so a cannot be 0 1 so hence you can cancel what r to the power n minus 1 also yep divide the entire equation by r to the power n minus 1 this will give you what 1 is equal to r plus r squared so hence your quadratic equation is r square plus r minus 1 is 0 right so hence r is minus b plus i will not take minus because it's a positive GP terms terms are all positive so root over b square that is 1 minus 4 ac that is plus 4 divided by 2 so minus 1 plus this is r combination easy no problem