A block of mass ' m ' (as shown in figure) moving with kinetic...

Using work $-$ energy theorem

${W}_{net} = ∆ KE$

${W}_{net} = \left({K}_{f} - {K}_{i}\right)$

$⇒ - \dfrac{1}{2}K{x}^{2} = \dfrac{1}{2}m{\left(\dfrac{v}{2}\right)}^{2} - \dfrac{1}{2}m{v}^{2} ⇒ - \dfrac{1}{2}K{x}^{2} = \dfrac{E}{4} - E$

$⇒ \dfrac{1}{2}K{x}^{2} = \dfrac{3E}{4} ⇒ K = \dfrac{3E}{2{x}^{2}}$

$⇒ K = \dfrac{3E}{2 \times {\left(\dfrac{1}{4}\right)}^{2}} = 24EN{m}^{- 1}$

Thus, the value of $n = 24$.