Explanation

(a) Position of $n$th bright fringe from central maxima
$x_{n1} = \frac{n_{1}\lambda D}{d}$ here $n_{1} = 5\ $
$\therefore x_{n1} = \frac{5\lambda D}{d}$
Position of n th dark fringe from central maxima
$x_{n} = \frac{(2n - 1)\lambda D}{2d},\ \ $here $n = 3$
$x_{n} = \frac{5}{2}\frac{\lambda D}{d}$
$x_{n1} - x_{n} = \frac{2.5\lambda D}{d} = 2.5\ \beta$
Given $\beta = 0.4\ mm$
$\Rightarrow x_{n1} - x_{n} = 1\ mm$