If `U_(n) = (1)/( n) - (1)/( n + 1) ` , then the value of ` U_(1) + U_(2) + U_(3) + U_(4) + U_(5)` is

Options

A. `(1)/(4)`
B. `(5)/( 6)`
C. `(4)/(5)`
D. `(1)/(3)`

Answer

To solve the problem, we need to calculate the sum \( U(1) + U(2) + U(3) + U(4) + U(5) \) where \( U(n) = \frac{1}{n} - \frac{1}{n+1} \).

### Step-by-Step Solution:

1. **Calculate \( U(1) \)**:
\[ U(1) = \frac{1}{1} - \frac{1}{1+1} = 1 - \frac{1}{2} = \frac{1}{2} \]

2. **Calculate \( U(2) \)**:
\[ U(2) = \frac{1}{2} - \frac{1}{2+1} = \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

3. **Calculate \( U(3) \)**:
\[ U(3) = \frac{1}{3} - \frac{1}{3+1} = \frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12} \]

4. **Calculate \( U(4) \)**:
\[ U(4) = \frac{1}{4} - \frac{1}{4+1} = \frac{1}{4} - \frac{1}{5} = \frac{5}{20} - \frac{4}{20} = \frac{1}{20} \]

5. **Calculate \( U(5) \)**:
\[ U(5) = \frac{1}{5} - \frac{1}{5+1} = \frac{1}{5} - \frac{1}{6} = \frac{6}{30} - \frac{5}{30} = \frac{1}{30} \]

6. **Sum all the values**:
\[ U(1) + U(2) + U(3) + U(4) + U(5) = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of \( 2, 6, 12, 20, 30 \) is \( 60 \).

- Convert each fraction:
\[ \frac{1}{2} = \frac{30}{60}, \quad \frac{1}{6} = \frac{10}{60}, \quad \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]

- Now sum them:
\[ U(1) + U(2) + U(3) + U(4) + U(5) = \frac{30}{60} + \frac{10}{60} + \frac{5}{60} + \frac{3}{60} + \frac{2}{60} = \frac{50}{60} = \frac{5}{6} \]

### Final Answer:
\[ U(1) + U(2) + U(3) + U(4) + U(5) = \frac{5}{6} \]