Question

If $a^2(b+c),b^2(c+a),c^2(a+b)$ are in A.P. , then $a,b,c$ are in

• A. A.P
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is A A.P

$a^2(b+c),b^2(c+a),c^2(a+b)$ are in A.P

$\Rightarrow 2b^2(c+a)=a^2(b+c)+c^2(a+b)$

$\Rightarrow 2b^{2}c+2b^{2}a=a^{2}b+a^{2}c+c^{2}a+c^{2}b$

$\Rightarrow b^{2}c+b^{2}c+b^{2}a+b^{2}a-a^{2}b-a^{2}c-c^{2}a-c^{2}b=0$

$\Rightarrow c(b^2-a^2)+ab(b-a)-a(c^2-b^2)-bc(c-b)$

$\Rightarrow c(b^2-a^2)+ab(b-a)=a(c^2-b^2)+bc(c-b)$

$\Rightarrow (b-a)[bc+ac+ab]=(c-b)[ac+ab+bc]$

$\Rightarrow b-a=c-b$

$\therefore$ $a,b,c$ are in $A.P$