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 find the value of the expression \( \frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143} \), we can follow these steps:

### Step 1: Rewrite each term
We can express each denominator as a product of two consecutive odd numbers:
- \( 15 = 3 \times 5 \)
- \( 35 = 5 \times 7 \)
- \( 63 = 7 \times 9 \)
- \( 99 = 9 \times 11 \)
- \( 143 = 11 \times 13 \)

So we rewrite the expression as:
\[ \frac{1}{3 \times 5} + \frac{1}{5 \times 7} + \frac{1}{7 \times 9} + \frac{1}{9 \times 11} + \frac{1}{11 \times 13} \]

### Step 2: Use the formula for partial fractions
We can use the partial fraction decomposition:
\[ \frac{1}{a(b)} = \frac{1}{b-a} - \frac{1}{b} \]
Applying this to each term:
- \( \frac{1}{3 \times 5} = \frac{1}{3} - \frac{1}{5} \)
- \( \frac{1}{5 \times 7} = \frac{1}{5} - \frac{1}{7} \)
- \( \frac{1}{7 \times 9} = \frac{1}{7} - \frac{1}{9} \)
- \( \frac{1}{9 \times 11} = \frac{1}{9} - \frac{1}{11} \)
- \( \frac{1}{11 \times 13} = \frac{1}{11} - \frac{1}{13} \)

### Step 3: Combine the terms
Now, substituting back into the expression:
\[ \left(\frac{1}{3} - \frac{1}{5}\right) + \left(\frac{1}{5} - \frac{1}{7}\right) + \left(\frac{1}{7} - \frac{1}{9}\right) + \left(\frac{1}{9} - \frac{1}{11}\right) + \left(\frac{1}{11} - \frac{1}{13}\right) \]

Notice that all intermediate terms cancel out:
\[ \frac{1}{3} - \frac{1}{13} \]

### Step 4: Simplify the remaining terms
Now we simplify:
\[ \frac{1}{3} - \frac{1}{13} = \frac{13 - 3}{39} = \frac{10}{39} \]

### Step 5: Final result
Thus, the value of the expression \( \frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143} \) is:
\[ \frac{10}{39} \]