The coefficient of the `(2m+1)^("th")` and `(4m+5)^("th")` terms in the expansion of `(1+x)^(100)` are equal, then the value of `(m)/(2)` is equal to

Answer

To solve the problem, we are tasked with finding the value of \( \frac{m}{2} \) given that the coefficients of the \( (2m + 1)^{\text{th}} \) and \( (4m + 5)^{\text{th}} \) terms in the expansion of \( (1+x)^{100} \) are equal.

### Step-by-step Solution:

1. **Understanding the General Term**:
The general term in the binomial expansion of \( (1+x)^{n} \) is given by:
\[ T_r = \binom{n}{r} x^r \]
where \( n \) is the exponent, and \( r \) is the term number starting from 0.

2. **Identifying the Coefficients**:
In our case, we need the coefficients of the \( (2m + 1)^{\text{th}} \) and \( (4m + 5)^{\text{th}} \) terms. The coefficients can be expressed as:
\[ \text{Coefficient of } (2m + 1)^{\text{th}} \text{ term} = \binom{100}{2m} \]
\[ \text{Coefficient of } (4m + 5)^{\text{th}} \text{ term} = \binom{100}{4m + 5} \]

3. **Setting Up the Equation**:
Since the coefficients are equal, we have:
\[ \binom{100}{2m} = \binom{100}{4m + 5} \]

4. **Using the Property of Binomial Coefficients**:
The property of binomial coefficients states that:
\[ \binom{n}{r} = \binom{n}{n-r} \]
Therefore, we can write:
\[ \binom{100}{4m + 5} = \binom{100}{100 - (4m + 5)} = \binom{100}{95 - 4m} \]
This leads us to the equation:
\[ 2m = 95 - 4m \]

5. **Solving for \( m \)**:
Rearranging the equation gives:
\[ 2m + 4m = 95 \]
\[ 6m = 95 \]
\[ m = \frac{95}{6} \]

6. **Finding \( \frac{m}{2} \)**:
Now, we need to find \( \frac{m}{2} \):
\[ \frac{m}{2} = \frac{95}{6} \cdot \frac{1}{2} = \frac{95}{12} \]

### Final Answer:
Thus, the value of \( \frac{m}{2} \) is:
\[ \frac{m}{2} = \frac{95}{12} \]