A civil engineer needs to calculate the square root of a given number with high precision for designing a bridge. However, the engineer's calculator lacks a square root function. It is of interest to develop a numerical method to approximate the square root of a given number.

$($a$)$ Propose an iterative method for approximating the square root using the Newton$-$Raphson method. $[5$ marks$]$

$($b$)$ Apply your proposed method to approximate the square root of $25.$ Start with an initial guess $x_0=5$ and iterate until the difference between successive approximations is less than ${10}^{-6}.$ Do not use a programming code to implement your method. $[10$ marks$]$