`1/(1.2.3)+1/(2.3.4)+1/(3.4.5)+1/(4.5.6)`is equal to

Options

A. A)(7/30)
B. B)(11/30)
C. C)(13/30)
D. D)(17/30)

Answer

To solve the expression \( \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{3 \cdot 4 \cdot 5} + \frac{1}{4 \cdot 5 \cdot 6} \), we will follow these steps:

### Step 1: Calculate each term's denominator
We first calculate the denominators for each term:
- \( 1 \cdot 2 \cdot 3 = 6 \)
- \( 2 \cdot 3 \cdot 4 = 24 \)
- \( 3 \cdot 4 \cdot 5 = 60 \)
- \( 4 \cdot 5 \cdot 6 = 120 \)

### Step 2: Rewrite the expression
Now we can rewrite the expression with the calculated denominators:
\[ \frac{1}{6} + \frac{1}{24} + \frac{1}{60} + \frac{1}{120} \]

### Step 3: Find the least common multiple (LCM)
Next, we find the least common multiple (LCM) of the denominators \( 6, 24, 60, \) and \( 120 \). The LCM is \( 120 \).

### Step 4: Rewrite each fraction with the LCM as the denominator
Now we convert each fraction to have a denominator of \( 120 \):
- For \( \frac{1}{6} \):
\[ \frac{1}{6} = \frac{20}{120} \quad \text{(since \( 120 \div 6 = 20 \))} \]
- For \( \frac{1}{24} \):
\[ \frac{1}{24} = \frac{5}{120} \quad \text{(since \( 120 \div 24 = 5 \))} \]
- For \( \frac{1}{60} \):
\[ \frac{1}{60} = \frac{2}{120} \quad \text{(since \( 120 \div 60 = 2 \))} \]
- For \( \frac{1}{120} \):
\[ \frac{1}{120} = \frac{1}{120} \]

### Step 5: Combine the fractions
Now we can combine all the fractions:
\[ \frac{20}{120} + \frac{5}{120} + \frac{2}{120} + \frac{1}{120} = \frac{20 + 5 + 2 + 1}{120} = \frac{28}{120} \]

### Step 6: Simplify the fraction
Now we simplify \( \frac{28}{120} \):
- The greatest common divisor (GCD) of \( 28 \) and \( 120 \) is \( 4 \).
- Dividing both the numerator and the denominator by \( 4 \):
\[ \frac{28 \div 4}{120 \div 4} = \frac{7}{30} \]

### Final Answer
Thus, the final answer is:
\[ \frac{7}{30} \]