Three numbers are in an increasing geometric progression with common ratio r . If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d . If the fourth term of GP is 3 r 2 , then r 2 - d is equal to :

Step-by-step solution

Three numbers are in an increasing geometric progression with common ratio r . Let first term is a ⇒ a r , a , a r   ∈   G . P Given that if middle term is doubled ⇒ a r , 2 a , a r   ∈   A . P If a ,   b ,   c   ∈   A .   P   ⇒   2 b   = a   +   c 4 a = a r + a r 4 = r + 1 r r 2 + 1 = 4 r r 2 - 4 r + 1 = 0 If a x 2   +   b x   + c   =   0   ⇒   x   =   - b &#1