The Newton-Raphson method provides an iterative approach to finding roots of a function. The formula for the Newton-Raphson method is:
\[
x(n + 1) = x(n) - \frac{f(x(n))}{f'(x(n))}
\]
For the function \( f(x) = x^2 - 153 \), the derivative is:
\[
f'(x) = 2x
\]
Substituting into the Newton-Raphson formula:
\[
x(n + 1) = x(n) - \frac{x(n)^2 - 153}{2x(n)}
\]
Simplifying the formula:
\[
x(n + 1) = \frac{2x(n)^2 - (x(n)^2 - 153)}{2x(n)} = \frac{x(n)^2 + 153}{2x(n)}
\]
Thus, the iterative formula is **\( x(n + 1) = 0.5 \left[ x(n) + \frac{153}{x(n)} \right] \)**.