Mathematics - Sequence & Series Question with Solution | TestHub

Put $u=1/a,v=1/b,w=1/c$, so $u+w=2v$. For A and B, reciprocate and subtract a common constant from $(a+b+c)/a,(a+b+c)/b,(a+b+c)/c$. For C, the reciprocal of $1/a+1/(b+c)$ is $a(b+c)/(a+b+c)$; the three numerators form an A.P. since $bc,ca,ab$ do. For D, put $U=1/\sqrt{a},V=1/\sqrt{b},W=1/\sqrt{c}$. The three terms are $1/(V+W),1/(W+U),1/(U+V)$. Their A.P. residual has numerator $U^2+W^2-2V^2=1/a+1/c-2/b=0$.