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 need to find the value of \( \frac{m}{2} \) given that the coefficients of the \( (2m + 1)^{th} \) and \( (4m + 5)^{th} \) terms in the expansion of \( (1 + x)^{100} \) are equal.

### Step-by-step Solution:

1. **Identify the General Term**:
The general term \( T_r \) in the expansion of \( (1 + x)^{100} \) is given by:
\[ T_r = \binom{100}{r} x^r \]
where \( r \) is the term number starting from 0.

2. **Find the Coefficient of the \( (2m + 1)^{th} \) Term**:
For the \( (2m + 1)^{th} \) term, we have:
\[ r = 2m \]
Thus, the coefficient is:
\[ \text{Coefficient of } T_{2m} = \binom{100}{2m} \]

3. **Find the Coefficient of the \( (4m + 5)^{th} \) Term**:
For the \( (4m + 5)^{th} \) term, we have:
\[ r = 4m + 4 \]
Thus, the coefficient is:
\[ \text{Coefficient of } T_{4m + 4} = \binom{100}{4m + 4} \]

4. **Set the Coefficients Equal**:
According to the problem, these coefficients are equal:
\[ \binom{100}{2m} = \binom{100}{4m + 4} \]

5. **Apply the Property of Binomial Coefficients**:
The equality of binomial coefficients gives us two cases:
- Case 1: \( 2m = 4m + 4 \)
- Case 2: \( 2m + (4m + 4) = 100 \)

6. **Solve Case 1**:
From \( 2m = 4m + 4 \):
\[ 2m - 4m = 4 \implies -2m = 4 \implies m = -2 \]
(This value is not valid since \( m \) must be non-negative.)

7. **Solve Case 2**:
From \( 2m + 4m + 4 = 100 \):
\[ 6m + 4 = 100 \implies 6m = 96 \implies m = 16 \]

8. **Find \( \frac{m}{2} \)**:
Now, we need to find \( \frac{m}{2} \):
\[ \frac{m}{2} = \frac{16}{2} = 8 \]

### Final Answer:
The value of \( \frac{m}{2} \) is \( 8 \).