If two consecutive terms in the expansion of `(x+a)^n` are equal to where n is a positive integer then `((n+1)a)/(x+a)` is

Options

A. Negative integer
B. rational number
C. a real number
D. a positive integer

Answer

To solve the problem, we need to analyze the condition given about the consecutive terms in the expansion of \((x + a)^n\).

### Step-by-Step Solution:

1. **Identify the Terms**:
The \(r^{th}\) term (\(T_r\)) and the \((r+1)^{th}\) term (\(T_{r+1}\)) in the expansion of \((x + a)^n\) can be expressed using the binomial theorem:
\[ T_r = \binom{n}{r-1} x^{n-(r-1)} a^{r-1} = \binom{n}{r-1} x^{n-r+1} a^{r-1} \]
\[ T_{r+1} = \binom{n}{r} x^{n-r} a^r \]

2. **Set the Terms Equal**:
According to the problem, these two consecutive terms are equal:
\[ T_r = T_{r+1} \]
This gives us the equation:
\[ \binom{n}{r-1} x^{n-r+1} a^{r-1} = \binom{n}{r} x^{n-r} a^r \]

3. **Simplify the Equation**:
We can simplify this equation. Dividing both sides by \(x^{n-r}\) and \(a^{r-1}\) (assuming \(x \neq 0\) and \(a \neq 0\)):
\[ \binom{n}{r-1} \frac{x}{a} = \binom{n}{r} \]

4. **Use the Binomial Coefficient Identity**:
We know that:
\[ \binom{n}{r} = \frac{n - r + 1}{r} \binom{n}{r-1} \]
Substituting this into our equation gives:
\[ \binom{n}{r-1} \frac{x}{a} = \frac{n - r + 1}{r} \binom{n}{r-1} \]
Assuming \(\binom{n}{r-1} \neq 0\), we can cancel it out:
\[ \frac{x}{a} = \frac{n - r + 1}{r} \]

5. **Rearranging the Equation**:
Rearranging gives us:
\[ x = a \cdot \frac{n - r + 1}{r} \]

6. **Finding the Value of \(\frac{(n + 1)a}{x + a}\)**:
We need to find:
\[ \frac{(n + 1)a}{x + a} \]
Substituting \(x\) from the previous step:
\[ x + a = a \cdot \frac{n - r + 1}{r} + a = a \left(\frac{n - r + 1 + r}{r}\right) = a \left(\frac{n + 1}{r}\right) \]
Therefore:
\[ \frac{(n + 1)a}{x + a} = \frac{(n + 1)a}{a \left(\frac{n + 1}{r}\right)} = r \]

7. **Conclusion**:
Since \(r\) is a term number, it is a positive integer. Therefore, \(\frac{(n + 1)a}{x + a}\) is a positive integer.

### Final Answer:
\[ \frac{(n + 1)a}{x + a} \text{ is a positive integer.} \]