If x, y, z are in arithmetic progression with common difference d,...

Given $x, y, z$ are in arithmetic progression, hence $2y = x + z,$ also \left|$\begin{cases}3 & 4\sqrt{2} & x \\ 4 & 5\sqrt{2} & y \\ 5 & k & z\end{cases}$\right| = 0$</p><p>Applying${R}_{2} \to {R}_{1} + {R}_{3} - 2{R}_{2}$</p><p>$⇒ \left|$\begin{cases}3 & 4\sqrt{2} & x \\ 0 & k - 6\sqrt{2} & 0 \\ 5 & k & z\end{cases}$\right| = 0$</p><p>$⇒ ( k - 6\sqrt{2} ) ( 3z - 5x ) = 0$</p><p>$⇒ ( k - 6\sqrt{2} ) = 0 or ( 3z - 5x ) = 0

If $3z - 5x = 0$

$⇒ 3 ( x + 2d ) - 5x = 0$

$⇒ x = 3d$ (Not possible)

$⇒ k = 6\sqrt{2}$

$⇒ {k}^{2} = 72 .</p>$