Question 1207627
If a,b,c are in AP then prove that a^2(b+c),b^2(c+a),c^2(a+b) are in AP.
<pre>
let the common difference be d

a = b-d
c = b+d

{{{matrix(1,5,a^2(b+c),",",b^2(c+a),",",c^2(a+b))}}}

{{{matrix(1,5,(b-d)^2(b+(b+d)^""),",",b^2((b+d)^""+(b-d)),",",(b+d)^2((b-d)^""+b))}}}

{{{matrix(1,5,(b-d)^2(2b+d),",",b^2(2b),",",(b+d)^2(2b-d))}}}

{{{matrix(1,5,(b^2-2bd+d^2)(2b+d),",",2b^3,",",(b^2+2bd+d^2)(2b-d))}}}

{{{matrix(1,5,2b^3 - 3b^2d + d^3,",",2b^3,",",2b^3 + 3b^2d - d^3)}}}

{{{matrix(1,5,2b^3 - (3b^2d - d^3),",",2b^3,",",2b^3 + (3 b^2d - d^3))}}}

So they are in AP with common difference {{{(3b^2d - d^3)}}}

Edwin</pre>