The expansion `(a+x)^(n) = .^(n)c_(0)a^(n) + ^(n)c_(1)a^(n-1)x +` ………… +` .^(n)c_(n)x^(n)` is valid when n is

Options

A. An integer
B. A rational number
C. An irrational number
D. A natural number

Answer

To determine when the expansion \((a+x)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} x^{k}\) is valid, we need to analyze the conditions under which the binomial theorem applies.

### Step-by-Step Solution:

1. **Understanding the Binomial Theorem**:
The binomial theorem states that for any real numbers \(a\) and \(x\), and any integer \(n\), the expression \((a + x)^{n}\) can be expanded as:
\[ (a + x)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} x^{k} \]
where \(\binom{n}{k}\) is the binomial coefficient.

2. **Conditions for \(n\)**:
The binomial coefficients \(\binom{n}{k}\) are defined only when \(n\) is a non-negative integer. This is because:
- The binomial coefficient \(\binom{n}{k}\) is defined as:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
- For \(k\) to be a valid index (where \(0 \leq k \leq n\)), \(n\) must be a non-negative integer.

3. **Conclusion**:
Therefore, the expansion \((a+x)^{n}\) is valid when \(n\) is a natural number (which includes all non-negative integers: \(0, 1, 2, 3, \ldots\)).

Thus, the answer is:
- The expansion is valid when \(n\) is a natural number.