Of a,b,c are in A.P., Prove that the following is also in A.P. : a^2(b+c),b^2(c+a),c^2(a+b)

Dear Student
Given, a,b and c are in A.P.

a²(b + c), b²(c + a), c²(a + b) will be in A.P.

if b²(c + a) – a²(b + c) = c²(a + b) – b²(c + a)

i.e., if c(b² – a²) + ab(b – a) = a(c² – b²) + bc(c – b)

i.e., if (b – a)(ab + bc + ca) = (c – b)(ab + bc + ca)

i.e., if b – a = c – b

i.e., if 2b = a + c

i.e., if a, b, c are in A.P.

Thus, if a, b, c are in A.P. then a²(b + c), b²(c + a), c²(a + b) will also be in A.P.

Regards