`(1 + x+x^2)^8 = a_0 + a_1x +…….+a_16 x^16` then `a_1 - a_3 + a_5- a_7 + ……..-a_15= `

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the value of \( a_1 - a_3 + a_5 - a_7 + \ldots - a_{15} \) from the expansion of \( (1 + x + x^2)^8 \).

### Step 1: Set up the equation
We start with the expression:
\[ (1 + x + x^2)^8 = a_0 + a_1 x + a_2 x^2 + \ldots + a_{16} x^{16} \]
This means that the coefficients \( a_i \) represent the coefficients of \( x^i \) in the expansion.

### Step 2: Substitute \( x \) with \( -x \)
Now, we will substitute \( x \) with \( -x \):
\[ (1 - x + x^2)^8 = a_0 - a_1 x + a_2 x^2 - a_3 x^3 + \ldots + (-1)^{16} a_{16} x^{16} \]

### Step 3: Write the two equations
From the original equation, we have:
\[ (1 + x + x^2)^8 = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots + a_{16} x^{16} \quad \text{(Equation 1)} \]
From the substituted equation, we have:
\[ (1 - x + x^2)^8 = a_0 - a_1 x + a_2 x^2 - a_3 x^3 + \ldots + a_{16} x^{16} \quad \text{(Equation 2)} \]

### Step 4: Subtract Equation 1 from Equation 2
Now, we subtract Equation 1 from Equation 2:
\[ (1 - x + x^2)^8 - (1 + x + x^2)^8 = (-2a_1 x - 2a_3 x^3 - 2a_5 x^5 - \ldots - 2a_{15} x^{15}) \]
This simplifies to:
\[ \frac{(1 - x + x^2)^8 - (1 + x + x^2)^8}{-2x} = a_1 + a_3 x^2 + a_5 x^4 + \ldots + a_{15} x^{14} \]

### Step 5: Evaluate at \( x = i \)
Next, we substitute \( x = i \) (where \( i = \sqrt{-1} \)):
\[ (1 + i + i^2)^8 - (1 - i + i^2)^8 = 0 \]
Since \( i^2 = -1 \), we have:
\[ (1 + i - 1)^8 - (1 - i - 1)^8 = (i)^8 - (-i)^8 \]
Calculating \( i^8 = 1 \) and \( (-i)^8 = 1 \):
\[ 1 - 1 = 0 \]

### Step 6: Conclusion
Thus, we find that:
\[ a_1 - a_3 + a_5 - a_7 + \ldots - a_{15} = 0 \]

### Final Answer
The value of \( a_1 - a_3 + a_5 - a_7 + \ldots - a_{15} \) is:
\[ \boxed{0} \]