If n is a positive integer, then the number of terms in the expansion of `(x+a)^n` is

Options

A. n+1
B. n-1
C. n
D. `n^(2)`

Answer

To find the number of terms in the expansion of \((x + a)^n\), where \(n\) is a positive integer, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Binomial Expansion**:
The binomial theorem states that:
\[ (x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k \]
Here, \(\binom{n}{k}\) is the binomial coefficient, which gives the number of ways to choose \(k\) elements from \(n\).

2. **Identifying the Range of \(k\)**:
In the expansion, \(k\) takes values from \(0\) to \(n\). This means that \(k\) can be \(0, 1, 2, \ldots, n\).

3. **Counting the Terms**:
The total number of different values that \(k\) can take is \(n - 0 + 1\). This is because we include both endpoints (0 and \(n\)).
\[ \text{Number of terms} = n - 0 + 1 = n + 1 \]

4. **Conclusion**:
Therefore, the number of terms in the expansion of \((x + a)^n\) is \(n + 1\).

### Final Answer:
The number of terms in the expansion of \((x + a)^n\) is \(n + 1\).

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