If `(1 + x + x^(2) + x^(3))^(n)= a_(0) + a_(1)x + a_(2)x^(2) + a_(3) x^(3) +...+ a_(3n) x^(3n)`, then the
value of `a_(0) + a_(4) +a_(8) + a_(12)+….. ` is

Options

A. -1
B. 0
C. `4^(n-1)`
D. n

Explanation

To find the value of \( a_0 + a_4 + a_8 + a_{12} + \ldots \) from the expression \( (1 + x + x^2 + x^3)^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots + a_{3n} x^{3n} \), we can follow these steps:

### Step 1: Simplify the expression
The expression \( 1 + x + x^2 + x^3 \) can be rewritten using the formula for the sum of a geometric series:
\[ 1 + x + x^2 + x^3 = \frac{1 - x^4}{1 - x} \quad \text{for } x \neq 1 \]
Thus, we have:
\[ (1 + x + x^2 + x^3)^n = \left( \frac{1 - x^4}{1 - x} \right)^n \]

### Step 2: Evaluate at specific values of \( x \)
To find the coefficients \( a_0, a_4, a_8, \ldots \), we will evaluate the expression at specific values of \( x \).

#### Step 2.1: Set \( x = 1 \)
\[ (1 + 1 + 1 + 1)^n = 4^n = a_0 + a_1 + a_2 + a_3 + \ldots + a_{3n} \]
This gives us:
\[ a_0 + a_1 + a_2 + a_3 + \ldots + a_{3n} = 4^n \]

#### Step 2.2: Set \( x = -1 \)
\[ (1 - 1 + 1 - 1)^n = 0^n = 0 = a_0 - a_1 + a_2 - a_3 + \ldots + (-1)^{3n} a_{3n} \]
This gives us:
\[ a_0 - a_1 + a_2 - a_3 + \ldots + (-1)^{3n} a_{3n} = 0 \]

#### Step 2.3: Set \( x = i \) (where \( i \) is the imaginary unit)
\[ (1 + i + i^2 + i^3)^n = (1 + i - 1 - i)^n = 0^n = 0 \]
This gives us:
\[ a_0 + a_1 i - a_2 - a_3 i = 0 \]

#### Step 2.4: Set \( x = -i \)
\[ (1 - i + (-i)^2 + (-i)^3)^n = (1 - i - 1 + i)^n = 0^n = 0 \]
This gives us:
\[ a_0 - a_1 i - a_2 + a_3 i = 0 \]

### Step 3: Add the equations
Now, we will add the equations obtained from \( x = 1 \), \( x = -1 \), \( x = i \), and \( x = -i \):

1. From \( x = 1 \):
\[ a_0 + a_1 + a_2 + a_3 + \ldots + a_{3n} = 4^n \]
2. From \( x = -1 \):
\[ a_0 - a_1 + a_2 - a_3 + \ldots + (-1)^{3n} a_{3n} = 0 \]
3. From \( x = i \):
\[ a_0 + a_1 i - a_2 - a_3 i = 0 \]
4. From \( x = -i \):
\[ a_0 - a_1 i - a_2 + a_3 i = 0 \]

### Step 4: Solve for the coefficients
By adding the equations from \( x = 1 \) and \( x = -1 \), we can isolate the terms that contribute to \( a_0 + a_4 + a_8 + \ldots \):
\[ 4^n + 0 = 2a_0 + 2a_2 + 2a_4 + \ldots \]
This simplifies to:
\[ a_0 + a_4 + a_8 + \ldots = \frac{4^n}{2} = 2^{2n - 1} \]

### Final Result
Thus, the value of \( a_0 + a_4 + a_8 + a_{12} + \ldots \) is:
\[ \boxed{2^{2n - 1}} \]