`((1)/(1.4)+(1)/(4.7)+(1)/(7.10)+(1)/(10.13)+(1)/(13.16))` is equal to

Options

A. A) `(1)/(3)`
B. B) `(5)/(16)`
C. C) `(3)/(8)`
D. D) `(41)/(7280)`

Answer

To solve the expression \(\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \frac{1}{10 \cdot 13} + \frac{1}{13 \cdot 16}\), we can simplify each term and look for a pattern.

### Step 1: Rewrite each term
We can rewrite each term in the expression:
\[ \frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \frac{1}{10 \cdot 13} + \frac{1}{13 \cdot 16} \]

### Step 2: Use the formula for partial fractions
Each term can be expressed using partial fractions:
\[ \frac{1}{a(a+b)} = \frac{1}{b} \left( \frac{1}{a} - \frac{1}{a+b} \right) \]
Applying this to each term:
- For \(\frac{1}{1 \cdot 4}\):
\[ \frac{1}{1 \cdot 4} = \frac{1}{3} \left( \frac{1}{1} - \frac{1}{4} \right) \]
- For \(\frac{1}{4 \cdot 7}\):
\[ \frac{1}{4 \cdot 7} = \frac{1}{3} \left( \frac{1}{4} - \frac{1}{7} \right) \]
- For \(\frac{1}{7 \cdot 10}\):
\[ \frac{1}{7 \cdot 10} = \frac{1}{3} \left( \frac{1}{7} - \frac{1}{10} \right) \]
- For \(\frac{1}{10 \cdot 13}\):
\[ \frac{1}{10 \cdot 13} = \frac{1}{3} \left( \frac{1}{10} - \frac{1}{13} \right) \]
- For \(\frac{1}{13 \cdot 16}\):
\[ \frac{1}{13 \cdot 16} = \frac{1}{3} \left( \frac{1}{13} - \frac{1}{16} \right) \]

### Step 3: Combine the terms
Now we can combine all these terms:
\[ \frac{1}{3} \left( \left( \frac{1}{1} - \frac{1}{4} \right) + \left( \frac{1}{4} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{10} \right) + \left( \frac{1}{10} - \frac{1}{13} \right) + \left( \frac{1}{13} - \frac{1}{16} \right) \right) \]

### Step 4: Simplify the expression
Notice that the intermediate terms cancel out:
\[ \frac{1}{3} \left( 1 - \frac{1}{16} \right) = \frac{1}{3} \left( \frac{16 - 1}{16} \right) = \frac{1}{3} \cdot \frac{15}{16} = \frac{15}{48} \]

### Step 5: Final simplification
Now we can simplify \(\frac{15}{48}\):
\[ \frac{15}{48} = \frac{5}{16} \]

Thus, the final answer is:
\[ \frac{5}{16} \]