The ratio of the coefficient of the middle term in the expansion of `(1+x)^(20)` and the sum of the coefficients of two middle terms in expansion of `(1+x)^(19)` is _____ .

Answer

To solve the problem, we need to find the ratio of the coefficient of the middle term in the expansion of \((1+x)^{20}\) and the sum of the coefficients of the two middle terms in the expansion of \((1+x)^{19}\).

### Step-by-Step Solution:

1. **Find the Coefficient of the Middle Term in \((1+x)^{20}\)**:
- The middle term in the expansion of \((1+x)^{20}\) can be found using the formula for the binomial expansion. Since \(n = 20\) (which is even), the middle term is given by:
\[ T_{10} = \binom{20}{10} x^{10} \]
- Therefore, the coefficient \(a\) of the middle term is:
\[ a = \binom{20}{10} \]

2. **Find the Two Middle Terms in \((1+x)^{19}\)**:
- For \((1+x)^{19}\), since \(n = 19\) (which is odd), the two middle terms are \(T_{9}\) and \(T_{10}\):
\[ T_{9} = \binom{19}{9} x^{9} \quad \text{and} \quad T_{10} = \binom{19}{10} x^{10} \]
- The coefficients of these terms are:
\[ b = \binom{19}{9} \quad \text{and} \quad c = \binom{19}{10} \]

3. **Sum of the Coefficients of the Two Middle Terms**:
- The sum of the coefficients \(b + c\) is:
\[ b + c = \binom{19}{9} + \binom{19}{10} \]
- By the binomial coefficient property, we know that:
\[ \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} \]
- Thus, we have:
\[ b + c = \binom{19}{9} + \binom{19}{10} = \binom{20}{10} \]

4. **Finding the Ratio**:
- Now, we can find the ratio of \(a\) to \(b + c\):
\[ \text{Ratio} = \frac{a}{b+c} = \frac{\binom{20}{10}}{\binom{20}{10}} = 1 \]

### Final Answer:
The ratio of the coefficient of the middle term in the expansion of \((1+x)^{20}\) and the sum of the coefficients of the two middle terms in the expansion of \((1+x)^{19}\) is **1**.