Suppose a, b, c are distinct real numbers. If a, b, c are in A.P. and a2 , b2 , c2 are in H.P., then

2b=a+c,b2=2a2c2a2+c2⇒ (a+c)24=2a2c2a2+c2⇒a2+c22+2aca2+c2−8a2c2=0⇒a2+c2+4aca2+c2−2ac=0⇒(a+c)2+2ac(a−c)2=0⇒(a+c)2+2ac=0 [∵ a≠c]⇒4b2+2ac=0⇒−a2,b,c are in G.P.