When simplified the sum `1/2+1/6+1/12+1/20+1/30+….+1/(n(n+1))` is equal to

Options

A. `1/n`
B. `1/(n+1)`
C. `(2(n-1))/n`
D. `n/(n+1)`

Answer

To simplify the sum \( S = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \ldots + \frac{1}{n(n+1)} \), we can express each term in the series in a different form.

### Step-by-Step Solution:

1. **Identify the general term**: The general term of the series can be expressed as:
\[ \frac{1}{n(n+1)} \]

2. **Use partial fractions**: We can decompose the term \( \frac{1}{n(n+1)} \) using partial fractions:
\[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \]

3. **Rewrite the sum**: Substituting the partial fraction decomposition into the sum, we have:
\[ S = \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \ldots + \left( \frac{1}{n} - \frac{1}{n+1} \right) \]

4. **Observe the telescoping nature**: Notice that this is a telescoping series. Most terms will cancel out:
\[ S = 1 - \frac{1}{n+1} \]

5. **Final simplification**: Therefore, the simplified form of the sum is:
\[ S = 1 - \frac{1}{n+1} = \frac{n}{n+1} \]

### Final Answer:
The sum \( S = \frac{n}{n+1} \).