Question

Prove that $\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$ = 0

Answer 1

Apply $R_1+R_3-2R_2$

$△=$ $\left|\begin{array}{ccc}0& 0& a+c-2b\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|$ = 0

Since a, b, c are in A.P.

$\therefore$ a+c=2b or a+c -2b=0

and as such $R_1$ is a row of zeros only.

$⟹△=0$