`((1)/(3.5) + (1)/(5.7) + (1)/(7.9) + (1)/(9.11) + (1)/(11.13) + (1)/(13.15))` is equal to

Options

A. `(2)/(45)`
B. `(4)/(45)`
C. `(7)/(45)`
D. `(2)/(15)`

Answer

To solve the expression

\[ \frac{1}{3.5} + \frac{1}{5.7} + \frac{1}{7.9} + \frac{1}{9.11} + \frac{1}{11.13} + \frac{1}{13.15}, \]

we will break it down step by step.

### Step 1: Rewrite the fractions
We can rewrite each term in the expression:

\[ \frac{1}{3.5} = \frac{1}{3 \times 5}, \quad \frac{1}{5.7} = \frac{1}{5 \times 7}, \quad \frac{1}{7.9} = \frac{1}{7 \times 9}, \]
\[ \frac{1}{9.11} = \frac{1}{9 \times 11}, \quad \frac{1}{11.13} = \frac{1}{11 \times 13}, \quad \frac{1}{13.15} = \frac{1}{13 \times 15}. \]

### Step 2: Combine the fractions
Now, we can express the entire sum:

\[ \frac{1}{3 \times 5} + \frac{1}{5 \times 7} + \frac{1}{7 \times 9} + \frac{1}{9 \times 11} + \frac{1}{11 \times 13} + \frac{1}{13 \times 15}. \]

### Step 3: Use the formula for partial fractions
We can use the identity:

\[ \frac{1}{a(b)} = \frac{1}{b} - \frac{1}{a} \]

to simplify each term.

For example,

\[ \frac{1}{3 \times 5} = \frac{1}{3} - \frac{1}{5}, \quad \frac{1}{5 \times 7} = \frac{1}{5} - \frac{1}{7}, \quad \frac{1}{7 \times 9} = \frac{1}{7} - \frac{1}{9}, \]
\[ \frac{1}{9 \times 11} = \frac{1}{9} - \frac{1}{11}, \quad \frac{1}{11 \times 13} = \frac{1}{11} - \frac{1}{13}, \quad \frac{1}{13 \times 15} = \frac{1}{13} - \frac{1}{15}. \]

### Step 4: Combine all the terms
Now, substituting back into the sum:

\[ \left(\frac{1}{3} - \frac{1}{5}\right) + \left(\frac{1}{5} - \frac{1}{7}\right) + \left(\frac{1}{7} - \frac{1}{9}\right) + \left(\frac{1}{9} - \frac{1}{11}\right) + \left(\frac{1}{11} - \frac{1}{13}\right) + \left(\frac{1}{13} - \frac{1}{15}\right). \]

### Step 5: Simplify the expression
Notice that all intermediate terms cancel out:

\[ \frac{1}{3} - \frac{1}{15}. \]

### Step 6: Calculate the final result
Now we need to find a common denominator to combine these two fractions. The common denominator of 3 and 15 is 15:

\[ \frac{1}{3} = \frac{5}{15}, \quad \frac{1}{15} = \frac{1}{15}. \]

So,

\[ \frac{5}{15} - \frac{1}{15} = \frac{4}{15}. \]

### Final Answer
Thus, the value of the expression

\[ \frac{1}{3.5} + \frac{1}{5.7} + \frac{1}{7.9} + \frac{1}{9.11} + \frac{1}{11.13} + \frac{1}{13.15} = \frac{4}{15}. \]