The value of `(1)/(15)+(1)/(35)+(1)/(63)+(1)/(99)+(1)/(143)` is

Options

A. `(5)/(39)`
B. `(4)/(39)`
C. `(2)/(39)`
D. `(7)/(39)`

Answer

To solve the problem \( \frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143} \), we can use a systematic approach.

### Step-by-Step Solution:

1. **Identify the Denominators**:
The denominators are \( 15, 35, 63, 99, \) and \( 143 \).

2. **Factor the Denominators**:
- \( 15 = 3 \times 5 \)
- \( 35 = 5 \times 7 \)
- \( 63 = 7 \times 9 \)
- \( 99 = 9 \times 11 \)
- \( 143 = 11 \times 13 \)

3. **Recognize the Pattern**:
Notice that each fraction can be expressed in terms of consecutive odd numbers:
- \( \frac{1}{15} = \frac{1}{3 \times 5} \)
- \( \frac{1}{35} = \frac{1}{5 \times 7} \)
- \( \frac{1}{63} = \frac{1}{7 \times 9} \)
- \( \frac{1}{99} = \frac{1}{9 \times 11} \)
- \( \frac{1}{143} = \frac{1}{11 \times 13} \)

4. **Use the Formula for Partial Fractions**:
The sum can be simplified using the formula:
\[ \frac{1}{n(n+2)} = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+2} \right) \]
Here, we can apply this to each term.

5. **Express Each Fraction**:
- \( \frac{1}{15} = \frac{1}{2} \left( \frac{1}{3} - \frac{1}{5} \right) \)
- \( \frac{1}{35} = \frac{1}{2} \left( \frac{1}{5} - \frac{1}{7} \right) \)
- \( \frac{1}{63} = \frac{1}{2} \left( \frac{1}{7} - \frac{1}{9} \right) \)
- \( \frac{1}{99} = \frac{1}{2} \left( \frac{1}{9} - \frac{1}{11} \right) \)
- \( \frac{1}{143} = \frac{1}{2} \left( \frac{1}{11} - \frac{1}{13} \right) \)

6. **Combine the Fractions**:
Now, adding these fractions together:
\[ \frac{1}{2} \left( \left( \frac{1}{3} - \frac{1}{5} \right) + \left( \frac{1}{5} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{9} \right) + \left( \frac{1}{9} - \frac{1}{11} \right) + \left( \frac{1}{11} - \frac{1}{13} \right) \right) \]
Notice that most terms cancel out:
\[ = \frac{1}{2} \left( \frac{1}{3} - \frac{1}{13} \right) \]

7. **Calculate the Remaining Terms**:
Now we need to calculate \( \frac{1}{3} - \frac{1}{13} \):
\[ \frac{1}{3} - \frac{1}{13} = \frac{13 - 3}{39} = \frac{10}{39} \]

8. **Final Calculation**:
Now multiply by \( \frac{1}{2} \):
\[ \frac{1}{2} \cdot \frac{10}{39} = \frac{10}{78} = \frac{5}{39} \]

Thus, the final answer is:
\[ \frac{5}{39} \]