When n is any postive integer,the expansion `(x+a)^(n) = .^(n)c_(0)x^(n)` + `.^(n)c_(1)x^(n-1)a` + ……. + `.^(n)c_(n)a^(n)` is valid only when

Options

A. `|x|lt1`
B. `|x|gt1`
C. `|x|lt1 and |a|lt1`
D. `x` and `a` are any two numbers

Answer

To solve the question regarding the validity of the binomial expansion \((x + a)^n\), we need to understand the conditions under which this expansion holds true.

### Step-by-Step Solution:

1. **Understanding the Binomial Theorem**:
The binomial theorem states that for any positive integer \(n\), the 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, defined as \(\frac{n!}{k!(n-k)!}\).

2. **Identifying the Conditions**:
The binomial expansion is valid under certain conditions. The primary condition is that \(n\) must be a non-negative integer (i.e., \(n \geq 0\)). This is because the binomial coefficients \(\binom{n}{k}\) are defined only for non-negative integers \(n\) and \(k\).

3. **Analyzing the Variables**:
The variables \(x\) and \(a\) can be any real numbers. There are no restrictions on \(x\) and \(a\) in the context of the binomial expansion. The only restriction is on \(n\).

4. **Conclusion**:
Therefore, the expansion \((x + a)^n\) is valid only when \(n\) is a positive integer (or more generally, a non-negative integer).

### Final Answer:
The expansion \((x + a)^n\) is valid only when \(n\) is a non-negative integer.