If the sum of odd terms and the sum of even terms in the expansion of `(x+a)^n` are p and q respectively then `p^2+q^2=`

Options

A. `((x+a)^(2n) - (x -a)^(2n))/(2)`
B. `(x + a)^(2n) - (x - a)^(2n)`
C. `((x + a)^(2n) + (x -a)^(2n))/(2)`
D. `(x + a)^(2n) + (x - a)^(2n)`

Answer

To solve the problem, we need to find the value of \( p^2 + q^2 \) given that \( p \) is the sum of the odd terms and \( q \) is the sum of the even terms in the expansion of \( (x + a)^n \).

### Step-by-Step Solution:

1. **Understanding the Binomial Expansion**:
The expansion of \( (x + a)^n \) is given by:
\[ (x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k \]
where \( \binom{n}{k} \) is the binomial coefficient.

2. **Identifying the Sums**:
- The sum of all terms \( S \) in the expansion is:
\[ S = (x + a)^n \]
- The sum of the even terms \( q \) can be expressed as:
\[ q = \sum_{k \text{ even}} \binom{n}{k} x^{n-k} a^k \]
- The sum of the odd terms \( p \) can be expressed as:
\[ p = \sum_{k \text{ odd}} \binom{n}{k} x^{n-k} a^k \]

3. **Using the Binomial Theorem**:
We can also express the sum of the even and odd terms using the expansions of \( (x + a)^n \) and \( (x - a)^n \):
\[ (x + a)^n + (x - a)^n = 2q \]
\[ (x + a)^n - (x - a)^n = 2p \]

4. **Finding \( p \) and \( q \)**:
From the above equations, we can express \( p \) and \( q \) as:
\[ q = \frac{(x + a)^n + (x - a)^n}{2} \]
\[ p = \frac{(x + a)^n - (x - a)^n}{2} \]

5. **Calculating \( p^2 + q^2 \)**:
We can use the identity:
\[ p^2 + q^2 = (p + q)^2 - 2pq \]
where:
\[ p + q = (x + a)^n \]
and:
\[ pq = \frac{(x + a)^n - (x - a)^n}{2} \cdot \frac{(x + a)^n + (x - a)^n}{2} \]

6. **Substituting Values**:
We can substitute the values of \( p + q \) and \( pq \) into the equation:
\[ p^2 + q^2 = \left( (x + a)^n \right)^2 - 2 \cdot \left( \frac{(x + a)^n - (x - a)^n}{2} \cdot \frac{(x + a)^n + (x - a)^n}{2} \right) \]

7. **Final Expression**:
After simplifying, we find:
\[ p^2 + q^2 = \frac{(x + a)^{2n} + (x - a)^{2n}}{2} \]

### Conclusion:
Thus, the value of \( p^2 + q^2 \) is:
\[ p^2 + q^2 = \frac{(x + a)^{2n} + (x - a)^{2n}}{2} \]