Institution: SONALI BANK LTD Exam: Sonali Bank Ltd - Assistant Programmer - 26.08.2016 ||
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To analyze the expression with the condition that a, b, and c are consecutive integers where \( a > b > c \), we can express them as \( a = n \), \( b = n-1 \), and \( c = n-2 \) for some integer \( n \).

Now, we evaluate each option:

1. **A) \( b + \frac{c}{a} = (n-1) + \frac{n-2}{n} = n - 1 + 1 - \frac{2}{n} = n - \frac{2}{n} \)**
2. **B) \( c + \frac{a}{b} = (n-2) + \frac{n}{n-1} = n - 2 + \frac{n}{n-1} \)** (which becomes complex)
3. **C) \( c + \frac{b}{a} = (n-2) + \frac{n-1}{n} = n - 2 + 1 - \frac{1}{n} = n - 1 - \frac{1}{n} \)**
4. **D) \( a + \frac{b}{c} = n + \frac{n-1}{n-2} = n + 1 - \frac{1}{n-2} \)**

To compare, as \( n \) increases, \( \frac{b}{c} \) approaches \( 1 \) because both b and c are close in value. The term \( n + (1 - \frac{1}{n-2}) \) will dominate as it increases linearly with \( n \). Thus, among the simplified expressions, option D consistently yields the highest value as \( n \) increases, making **D: \( a + \frac{b}{c} \)** the maximum value.