Question

If $a^2(b+c),b^2(c+a),c^2(a+b)$ are in AP, then $a,b,c$ are in

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

The correct option is A AP

Given: $a^2(b+c),b^2(c+a),c^2(a+b)$ are in AP

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

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

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

We know: $2b=a+c$

$\Rightarrow bc\left(a\right)+ab\left(c\right)=ac(a+c)$

$\Rightarrow 2abc=ac(a+c)$

$\Rightarrow 2b=a+c$

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