Question

if a b c are in ap then prove that a²(b+c) ,b²(c+a) , c²(a+b)​

Answer 1

∵ a, b, c are in AP

Hence, (b-a) = (c-b) ⟶ (1)

∵ b²(c+a) - a²(b+c)

= (b-a) (ab + bc + ca)

and c²(a+b) - b²(c+a)

= (c-b) (ab + bc + ca)

= (b−a) (ab + bc + ca) [from (1)]

Hence, b²(c+a) - a²(b+c) = c²(a+b) - b²(c+a)

∴ a²(b+c), b²(c+a), c²(a+b) are also in AP.