Let `f(n) = [1/2 + n/100]` where `[x]` denote the integral part of `x.` Then the value of `sum_(n=1)^100 f(n)` is

Options

A. 50
B. 51
C. 1
D. 52

Answer

To solve the problem, we need to evaluate the sum \( \sum_{n=1}^{100} f(n) \) where \( f(n) = \left[ \frac{1}{2} + \frac{n}{100} \right] \) and \( [x] \) denotes the integral part of \( x \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(n) \) is defined as the greatest integer less than or equal to \( \frac{1}{2} + \frac{n}{100} \). We need to find the values of \( f(n) \) for \( n = 1, 2, \ldots, 100 \).

2. **Calculate \( f(n) \)**:
- For \( n = 1 \):
\[ f(1) = \left[ \frac{1}{2} + \frac{1}{100} \right] = \left[ 0.5 + 0.01 \right] = \left[ 0.51 \right] = 0 \]
- For \( n = 2 \):
\[ f(2) = \left[ \frac{1}{2} + \frac{2}{100} \right] = \left[ 0.5 + 0.02 \right] = \left[ 0.52 \right] = 0 \]
- Continuing this way, we find that \( f(n) = 0 \) for \( n = 1, 2, \ldots, 49 \).

3. **Finding the Transition Point**:
- For \( n = 50 \):
\[ f(50) = \left[ \frac{1}{2} + \frac{50}{100} \right] = \left[ 0.5 + 0.5 \right] = \left[ 1 \right] = 1 \]
- For \( n = 51 \):
\[ f(51) = \left[ \frac{1}{2} + \frac{51}{100} \right] = \left[ 0.5 + 0.51 \right] = \left[ 1.01 \right] = 1 \]
- Continuing this way, we find that \( f(n) = 1 \) for \( n = 50, 51, \ldots, 100 \).

4. **Counting the Values**:
- From \( n = 1 \) to \( n = 49 \), \( f(n) = 0 \) (49 terms).
- From \( n = 50 \) to \( n = 100 \), \( f(n) = 1 \) (51 terms).

5. **Calculating the Sum**:
- The total sum can be calculated as:
\[ \sum_{n=1}^{100} f(n) = \sum_{n=1}^{49} f(n) + \sum_{n=50}^{100} f(n) = 0 \cdot 49 + 1 \cdot 51 = 51 \]

### Final Answer:
The value of \( \sum_{n=1}^{100} f(n) \) is \( \boxed{51} \).