Question

If a,b,c are in A.P then show that :-
a^2(b+c) , b^2(c+a) , c^2(a+b) are in A.P

Answer 1

@mathmate

Answer 2

The most direct way is to assume
a=a
b=a+d
c=a+2d
since a,b,c are in AP.
substitute the above into the three expressions, E1,E2,E3 respectively.
Calculate and check if E3-E2=E2-E1. If so, the three expressions are in AP.
I get E3-E2=E2-E1=2d^3+6ad^2+3a^2d.

Answer 3

But how to actually prove it

Answer 4

Without loss of generality, you can also assume a=0, which simplifies things a lot.

Answer 5

If a^2(b+c) , b^2(c+a) , c^2(a+b) are in A.P, then
they should have a common difference.
All you need to prove is that if a,b,c are in AP, then
c^2(a+b)-b^2(c+a) = b^2(c+a) - a^2(b+c) =D = common difference, hence
the three expressions are in AP.

Answer 6

For details, see initial post above.