If `f(x)={(2(1-x/2))^25(2+x)^25}^(1/50)` and `g(x)=f(f(f(x)))+f(f(x))` then find `[g(1)]` (where [x] is Greatest integer less than or equal to x)

Answer

To solve the problem, we need to evaluate the function \( g(x) = f(f(f(x))) + f(f(x)) \) at \( x = 1 \). Let's break this down step by step.

### Step 1: Calculate \( f(1) \)

The function \( f(x) \) is defined as:

\[ f(x) = \left( (2(1 - \frac{x}{2}))^{25} (2 + x)^{25} \right)^{\frac{1}{50}} \]

Substituting \( x = 1 \):

\[ f(1) = \left( (2(1 - \frac{1}{2}))^{25} (2 + 1)^{25} \right)^{\frac{1}{50}} \]

Calculating the terms:

\[ 1 - \frac{1}{2} = \frac{1}{2} \quad \text{and} \quad 2 + 1 = 3 \]

Thus,

\[ f(1) = \left( (2 \cdot \frac{1}{2})^{25} \cdot 3^{25} \right)^{\frac{1}{50}} = \left( 1^{25} \cdot 3^{25} \right)^{\frac{1}{50}} = \left( 3^{25} \right)^{\frac{1}{50}} = 3^{\frac{25}{50}} = 3^{\frac{1}{2}} = \sqrt{3} \]

### Step 2: Calculate \( f(f(1)) = f(\sqrt{3}) \)

Now, we need to find \( f(\sqrt{3}) \):

\[ f(\sqrt{3}) = \left( (2(1 - \frac{\sqrt{3}}{2}))^{25} (2 + \sqrt{3})^{25} \right)^{\frac{1}{50}} \]

Calculating the terms:

\[ 1 - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2} \quad \text{and} \quad 2 + \sqrt{3} \]

Thus,

\[ f(\sqrt{3}) = \left( \left( 2 \cdot \frac{2 - \sqrt{3}}{2} \right)^{25} (2 + \sqrt{3})^{25} \right)^{\frac{1}{50}} = \left( (2 - \sqrt{3})^{25} (2 + \sqrt{3})^{25} \right)^{\frac{1}{50}} \]

Using the identity \( (a-b)(a+b) = a^2 - b^2 \):

\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1 \]

Thus,

\[ f(\sqrt{3}) = \left( 1^{25} \right)^{\frac{1}{50}} = 1 \]

### Step 3: Calculate \( f(f(f(1))) = f(f(\sqrt{3})) = f(1) \)

Since we found \( f(\sqrt{3}) = 1 \):

\[ f(f(\sqrt{3})) = f(1) = \sqrt{3} \]

### Step 4: Calculate \( g(1) \)

Now we can calculate \( g(1) \):

\[ g(1) = f(f(f(1))) + f(f(1)) = f(1) + f(\sqrt{3}) = \sqrt{3} + 1 \]

### Step 5: Find the greatest integer less than or equal to \( g(1) \)

We know:

\[ \sqrt{3} \approx 1.732 \]

Thus,

\[ g(1) \approx 1.732 + 1 = 2.732 \]

Taking the greatest integer less than or equal to \( g(1) \):

\[ [g(1)] = 2 \]

### Final Answer

The final answer is:

\[ \boxed{2} \]