If a,b,c are three unequal numbers such that a, b, c are in A.P and b−a, c−b, a are in G.P, then a : b : c is

Question

If  a,b,c      are three unequal  numbers such that a, b, c      are in A.P and b−a,  c−b,  a      are in G.P, then a  :  b  :  c      is

Infinity Learn · Accepted Answer

Given  that a, b, c      are in A.P  ⇒            $$2b=a+c$$     and  b−a,  c−b,  a      are in G.P.⇒                (c$$−$$b)2=(b$$−$$a)a$$     ⇒   $$c2+b2$$−$$2bc=ab$$−$$a2$$     ⇒   $$c2+(a+c2)2$$−$$2(a+c2)c$$      $$=a(a+c2)−$$a2$$            [using  Eq.  (i)]     ⇒    $$c2+a2+c2+2ac4$$−ac−$$c2=a22+ac2$$−$$a2$$     ⇒   (c$$−$$a)2=2a(c$$−$$a)     ⇒               $$c=3a$$             $$($$∵   c≠$$a)$$     From  Equation.  (i)                $$2b=a+3a$$                       $$b=2a$$     ∴      a  :  b  :  $$c=a$$  :  $$2a$$  :  $$3a$$                            $$=1$$  :  $$2$$  :$$3$$

If a,b,c are three unequal numbers such that a, b, c are in A.P and b−a, c−b, a are in G.P, then a : b : c is

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