Question

Consider the following recursive iteration scheme for different values of variable P with the initial guess \( x_1 = 1 \):
\[ x_{n+1} = \frac{1}{2} \left( x_n + \frac{P}{x_n} \right), n = 1, 2, 3, 4, 5 \] For \( P = 2 \), \( x_5 \) is obtained to be 1.414, rounded-off to three decimal places. For \( P = 3 \), \( x_5 \) is obtained to be 1.732, rounded-off to three decimal places.
If \( P = 10 \), the numerical value of \( x_5 \) is \(\underline{\hspace{2cm}}\) (round off to three decimal places).

Hint:

To solve recursive iteration schemes, use the formula iteratively, updating \( x_n \) at each step.

Answer 1

Correct Answer: 3.1

Using the given recursive iteration scheme, we can calculate the values of \( x_2, x_3, x_4, \) and \( x_5 \) for \( P = 10 \). Starting with \( x_1 = 1 \), we perform the following iterations:
\[ x_2 = \frac{1}{2} \left( 1 + \frac{10}{1} \right) = 5.5 \] \[ x_3 = \frac{1}{2} \left( 5.5 + \frac{10}{5.5} \right) = 3.317 \] \[ x_4 = \frac{1}{2} \left( 3.317 + \frac{10}{3.317} \right) = 3.100 \] \[ x_5 = \frac{1}{2} \left( 3.100 + \frac{10}{3.100} \right) = 3.162. \] Thus, the numerical value of \( x_5 \) for \( P = 10 \) is \( 3.162 \).