Suppose the coefficient of the middle term in the expansion of `(1 + x)^(2n)` is A and the coefficients of two middle terms in the expansion of `(1 + x)^(2n-1)` are B and C, then `A/(B+C)=`_________

Answer

To solve the problem, we need to find the values of \( A \), \( B \), and \( C \) based on the coefficients of the middle terms in the expansions of \( (1 + x)^{2n} \) and \( (1 + x)^{2n-1} \).

### Step-by-Step Solution:

1. **Find the Coefficient \( A \)**:
- The middle term in the expansion of \( (1 + x)^{2n} \) can be found using the formula for the middle term.
- Since \( 2n \) is even, the middle term is given by the \( (n + 1) \)-th term.
- The general term \( T_{r+1} \) in the expansion is given by:
\[ T_{r+1} = \binom{2n}{r} x^r \]
- For the middle term \( T_{n+1} \) (where \( r = n \)):
\[ A = \binom{2n}{n} \]

2. **Find the Coefficients \( B \) and \( C \)**:
- Now consider the expansion of \( (1 + x)^{2n-1} \). Since \( 2n-1 \) is odd, there are two middle terms.
- The middle terms are given by the \( n \)-th and \( (n + 1) \)-th terms.
- The general term \( T_{r+1} \) in this expansion is:
\[ T_{r+1} = \binom{2n-1}{r} x^r \]
- For the \( n \)-th term:
\[ B = \binom{2n-1}{n-1} \]
- For the \( (n + 1) \)-th term:
\[ C = \binom{2n-1}{n} \]

3. **Calculate \( B + C \)**:
- Now, we need to find \( B + C \):
\[ B + C = \binom{2n-1}{n-1} + \binom{2n-1}{n} \]
- Using the property of binomial coefficients:
\[ \binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1} \]
- We have:
\[ B + C = \binom{2n-1}{n-1} + \binom{2n-1}{n} = \binom{2n}{n} \]

4. **Find the Ratio \( \frac{A}{B + C} \)**:
- Now substituting the values of \( A \) and \( B + C \):
\[ A = \binom{2n}{n} \]
\[ B + C = \binom{2n}{n} \]
- Therefore:
\[ \frac{A}{B + C} = \frac{\binom{2n}{n}}{\binom{2n}{n}} = 1 \]

### Final Answer:
\[ \frac{A}{B + C} = 1 \]