Question :

If a,b,c are in Geometric Progression , prove that (1/a+b)+(1/b+c) = (1/b) .

I found that b^2=ac , and thus c = (b^2)/a and a = (b^2)/c here .

I managed to solve this problem , by substituting "c" in the equation as , (b^2)/a , and simplifying , after which the left hand side finally equated to 1/b.

I ran into a doubt , why do I need to substitute only for "c" in the equation to equate ? Why cant one substitute for both "a" , "c" or "a" , "c" , "b" simultaneously ?

I found that , once I substitute for more than once variable in the equation , the equation no longer equates . This maybe a stupid doubt , but please help me clear that abstract gap in my mind .

Thanks so much .