`1/(5 xx 6) + 1/(6 xx 7) + 1/(7 xx 8) + ….. + 1/(24 xx 25)` ?

Options

A. 0.36
B. 0.16
C. 0.0016
D. 1.6

Answer

To solve the problem \( S = \frac{1}{5 \times 6} + \frac{1}{6 \times 7} + \frac{1}{7 \times 8} + \ldots + \frac{1}{24 \times 25} \), we can use the technique of partial fractions.

### Step-by-Step Solution:

1. **Rewrite Each Term**:
We can express each term in the series using the formula:
\[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \]
Therefore, we can rewrite the series:
\[ S = \left(\frac{1}{5} - \frac{1}{6}\right) + \left(\frac{1}{6} - \frac{1}{7}\right) + \left(\frac{1}{7} - \frac{1}{8}\right) + \ldots + \left(\frac{1}{24} - \frac{1}{25}\right) \]

2. **Combine the Terms**:
Notice that this is a telescoping series. Most terms will cancel out:
\[ S = \frac{1}{5} - \frac{1}{25} \]

3. **Calculate the Remaining Terms**:
Now we need to compute \( S \):
\[ S = \frac{1}{5} - \frac{1}{25} \]
To subtract these fractions, we need a common denominator, which is 25:
\[ S = \frac{5}{25} - \frac{1}{25} = \frac{5 - 1}{25} = \frac{4}{25} \]

4. **Convert to Decimal**:
Now, we can convert \( \frac{4}{25} \) into a decimal:
\[ \frac{4}{25} = 0.16 \]

### Final Answer:
Thus, the value of the series is:
\[ \boxed{0.16} \]