Let `f_(n)(theta) = sum_(n=0)^(n) (1)/(4^(n))sin^(4)(2^(n)theta)`. Then which of the following alternative(s) is/are correct ?

Options

A. `f_(2)((pi)/(4))=(pi)/(sqrt(2))`
B. `f_(3)((pi)/(8)) = (2 + sqrt(2))/(4)`
C. `f_(4)((3pi)/(2)) = 1`
D. `f_(5)(pi) = 0`

Explanation

To solve the problem, we need to analyze the function defined as:

\[ f_n(\theta) = \sum_{n=0}^{n} \frac{1}{4^n} \sin^4(2^n \theta) \]

### Step 1: Rewrite the sine function
We can rewrite \(\sin^4(x)\) using the identity \(\sin^4(x) = \left(\sin^2(x)\right)^2\) and the identity \(\sin^2(x) = 1 - \cos^2(x)\):

\[ \sin^4(2^n \theta) = \left(\sin^2(2^n \theta)\right)^2 = \left(1 - \cos^2(2^n \theta)\right)^2 \]

### Step 2: Substitute into the summation
Now, substituting this into our function:

\[ f_n(\theta) = \sum_{n=0}^{n} \frac{1}{4^n} \left(1 - \cos^2(2^n \theta)\right)^2 \]

### Step 3: Expand the square
Expanding the square gives:

\[ (1 - \cos^2(2^n \theta))^2 = 1 - 2\cos^2(2^n \theta) + \cos^4(2^n \theta) \]

### Step 4: Substitute back into the summation
Now substituting back into the summation:

\[ f_n(\theta) = \sum_{n=0}^{n} \frac{1}{4^n} \left(1 - 2\cos^2(2^n \theta) + \cos^4(2^n \theta)\right) \]

### Step 5: Separate the summation
We can separate the summation into three parts:

\[ f_n(\theta) = \sum_{n=0}^{n} \frac{1}{4^n} - 2 \sum_{n=0}^{n} \frac{\cos^2(2^n \theta)}{4^n} + \sum_{n=0}^{n} \frac{\cos^4(2^n \theta)}{4^n} \]

### Step 6: Evaluate the first summation
The first summation is a geometric series:

\[ \sum_{n=0}^{n} \frac{1}{4^n} = \frac{1}{1 - \frac{1}{4}} = \frac{4}{3} \]

### Step 7: Evaluate the second summation
The second summation can be evaluated using the known result for \(\sum_{n=0}^{\infty} x^n\):

\[ \sum_{n=0}^{n} \frac{\cos^2(2^n \theta)}{4^n} \]

This requires more advanced techniques, possibly involving generating functions or Fourier series, which may not yield a simple closed form.

### Step 8: Analyze the third summation
The third summation involves \(\cos^4(2^n \theta)\), which can also be evaluated using identities or numerical methods.

### Step 9: Final expression
Thus, the final expression for \(f_n(\theta)\) will depend on the evaluations of these summations.

### Conclusion
To determine which alternatives are correct, we would need to evaluate \(f_n(\theta)\) for specific values of \(\theta\) and \(n\) based on the derived expression.

---