The False Position Method (also called the Regula Falsi method) is a numerical method used to solve equations of the form $f(x) = 0$, where $f(x)$ is a continuous function. It iteratively refines the estimate of the root based on the values of $f(x)$ at two points that bracket the root.

Given equation:

$2e^{-x/2} + \ln(x) = 1$ We need to rewrite this equation in the form: $f(x) = 2e^{-x/2} + \ln(x) - 1 = 0$

We are also given the interval $(0.1, 1)$, which means the root lies between these two values. Let's now apply the False Position Method to generate the first four approximations of the root.

Step 1: Calculate $f(x)$ at the endpoints of the interval

We first evaluate the function $f(x)$ at the endpoints $x_0 = 0.1$ and $x_1 = 1$.

1. Evaluate $f(0.1)$:

$f(0.1) = 2e^{-0.1/2} + \ln(0.1) - 1 = 2e^{-0.05} + \ln(0.1) - 1$

2. Evaluate $f(1)$:

$f(1) = 2e^{-1/2} + \ln(1) - 1 = 2e^{-0.5} + 0 - 1$

Step 2: Calculate the first approximation $x_2$ using the False Position formula:

The False Position formula for the next approximation is: [ x_2 =