`1/3+1/15+1/35+1/63+1/99+1/143=?`

Options

A. a)`6/13`
B. b)`5/7`
C. c)`6/11`
D. d)`11/6`

Answer

To solve the problem \( \frac{1}{3} + \frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143} \), we can follow a systematic approach.

### Step-by-Step Solution:

1. **Identify the Denominators**:
The denominators are \( 3, 15, 35, 63, 99, \) and \( 143 \).

2. **Factor the Denominators**:
- \( 3 = 1 \times 3 \)
- \( 15 = 3 \times 5 \)
- \( 35 = 5 \times 7 \)
- \( 63 = 7 \times 9 \)
- \( 99 = 9 \times 11 \)
- \( 143 = 11 \times 13 \)

3. **Observe the Pattern**:
The denominators can be expressed in terms of consecutive odd numbers:
- \( 3 = 1 \times 3 \)
- \( 15 = 3 \times 5 \)
- \( 35 = 5 \times 7 \)
- \( 63 = 7 \times 9 \)
- \( 99 = 9 \times 11 \)
- \( 143 = 11 \times 13 \)

4. **Identify the Formula**:
We can use the formula for the sum of fractions of this type:
\[ \text{Sum} = \frac{1}{\text{difference of factors}} \times \left( \text{highest factor} - \text{lowest factor} \right) \]
Here, the difference of factors is \( 2 \) (since the difference between consecutive odd numbers is \( 2 \)), the lowest factor is \( 3 \), and the highest factor is \( 13 \).

5. **Apply the Formula**:
\[ \text{Sum} = \frac{1}{2} \times (13 - 1) = \frac{1}{2} \times 12 = \frac{12}{2} = 6 \]

6. **Final Calculation**:
Since we need to express the answer in terms of the sum of the fractions:
\[ \text{Final Answer} = \frac{6}{13} \]

### Conclusion:
Thus, the result of \( \frac{1}{3} + \frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143} \) is \( \frac{6}{13} \).