**Solution:**

Let's consider a positive real number *x*.

Step 1: Increase by 20%
The number is increased by 20%, which means it becomes *(x + 0.2x) = 1.2x*.

Step 2: Decrease by 25%
The number is decreased by 25%, which means it becomes *(1.2x - 0.25(1.2x)) = 0.9(1.2x) = 1.08x*.

Step 3: Increase by 10%
The number is increased by 10%, which means it becomes *(1.08x + 0.1(1.08x)) = 1.08(1.1)x = 1.188x*.

This process is repeated 49 more times, resulting in a new number each time. Let's denote the new number after *n* repetitions as *y(n)*.

**Recursive formula for *y(n)*:**
*y(n) = 1.188y(n-1)*

We can find the value of *y(n)* by applying this recursive formula repeatedly.

**Finding the value of *y(n)*:**

For *n = 1*, we have:
*y(1) = 1.188y(0)*

For *n = 2*, we have:
*y(2) = 1.188y(1) = 1.188(1.188y(0)) = (1.188)^2y(0)*

In general, for *n* repetitions, we have:
*y(n) = (1.188)^ny(0)*

So, the new number obtained after *n* repetitions is given by *(1.188)^ny(0)*.

**Analysis of the new number:**

The value of *(1.188)^n* is greater than 1 for all positive integers *n*. This means that the new number obtained after each repetition is greater than the previous number.

As *n* increases, the value of *(1.188)^n* increases exponentially. Therefore, the new number obtained after each repetition will also increase exponentially.

Since the initial number *x* is positive, the new number obtained after each repetition will also be positive.

In conclusion, the new number obtained after 49 repetitions will be a positive real number that is greater than the initial number *x*.