If the sum of odd numbered terms and the sum of even numbered terms in the expansion of `(x + a)^n` are A and B respectively, then the value of (`x^2 – a^2)^n` is

Options

A. `A^2 - B^2`
B. `A^2 + B^2`
C. `4AB`
D. None

Answer

To solve the problem, we need to find the value of \((x^2 - a^2)^n\) given the sums of the odd and even numbered terms in the expansion of \((x + a)^n\).

### Step-by-Step Solution:

1. **Understand the Expansion**:
The binomial expansion of \((x + a)^n\) can be expressed as:
\[ (x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k \]
where \(\binom{n}{k}\) is the binomial coefficient.

2. **Identify Odd and Even Terms**:
- The **even numbered terms** correspond to the terms where \(k\) is even (0, 2, 4, ...).
- The **odd numbered terms** correspond to the terms where \(k\) is odd (1, 3, 5, ...).

3. **Sum of Even and Odd Terms**:
Let \(A\) be the sum of the odd numbered terms and \(B\) be the sum of the even numbered terms. Thus, we can write:
\[ (x + a)^n = A + B \]

4. **Consider the Expansion of \((x - a)^n\)**:
The expansion of \((x - a)^n\) is:
\[ (x - a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} (-a)^k \]
This can be rewritten as:
\[ (x - a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k (-1)^k \]
Here, the even terms will remain positive and the odd terms will become negative.

5. **Sum of Terms for \((x - a)^n\)**:
- The sum of the even numbered terms remains the same as in \((x + a)^n\).
- The sum of the odd numbered terms will be the negative of the odd terms in \((x + a)^n\).
Thus:
\[ (x - a)^n = B - A \]

6. **Set Up the Equations**:
Now we have two equations:
\[ (x + a)^n = A + B \quad \text{(1)} \]
\[ (x - a)^n = B - A \quad \text{(2)} \]

7. **Add the Two Equations**:
Adding equations (1) and (2):
\[ (x + a)^n + (x - a)^n = (A + B) + (B - A) = 2B \]
Thus:
\[ 2B = (x + a)^n + (x - a)^n \]
Therefore:
\[ B = \frac{(x + a)^n + (x - a)^n}{2} \]

8. **Subtract the Two Equations**:
Subtracting equation (2) from equation (1):
\[ (x + a)^n - (x - a)^n = (A + B) - (B - A) = 2A \]
Thus:
\[ 2A = (x + a)^n - (x - a)^n \]
Therefore:
\[ A = \frac{(x + a)^n - (x - a)^n}{2} \]

9. **Find \((x^2 - a^2)^n\)**:
Now, we can find \((x^2 - a^2)^n\):
\[ (x^2 - a^2)^n = (x + a)^n (x - a)^n \]
Substituting the values of \(A\) and \(B\):
\[ (x^2 - a^2)^n = (A + B)(B - A) = A^2 - B^2 \]

10. **Final Result**:
Thus, the value of \((x^2 - a^2)^n\) can be expressed as:
\[ (x^2 - a^2)^n = A^2 - B^2 \]