Question

Show that $\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|=0\mathrm{where}a,b,c\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}.$

Answer 1

Given: a, b, c are in A.P.

$2b=a+c$

$$\begin{gathered} ∆=\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|\left[\mathrm{Applying}{R}_2=2R_2\right] \\ ∆=\frac{1}{2}\left|\begin{array}{ccc}x+1& x+2& x+a\\ 2x+4& 2x+6& 2x+2b\\ x+3& x+4& x+c\end{array}\right| \\ ∆=\frac{1}{2}\left|\begin{array}{ccc}x+1& x+2& x+a\\ 0& 0& 0\\ x+3& x+4& x+c\end{array}\right|\left[\because 2b=a+c\right]\left[\mathrm{Applying}{R}_2\to R_2-\left(R_1+R_3\right)\right] \\ ∆=0 \end{gathered}$$