Find the sum of first 20 terms of the sequence `1/(5xx6)+1/(6xx7)+1/(7xx8)+.....`

Options

A. 0.16
B. 1.6
C. 16
D. 0.016

Answer

To find the sum of the first 20 terms of the sequence given by \( S = \frac{1}{5 \times 6} + \frac{1}{6 \times 7} + \frac{1}{7 \times 8} + \ldots \), we can start by identifying the general term of the sequence.

### Step 1: Identify the general term
The \( n \)-th term of the sequence can be expressed as:
\[ a_n = \frac{1}{n(n+1)} \]
where \( n \) starts from 5. Therefore, the first term corresponds to \( n = 5 \), the second term corresponds to \( n = 6 \), and so on.

### Step 2: Write the first 20 terms
The first 20 terms will be:
\[ \frac{1}{5 \times 6}, \frac{1}{6 \times 7}, \frac{1}{7 \times 8}, \ldots, \frac{1}{24 \times 25} \]

### Step 3: Sum of the first 20 terms
We can use the formula for the sum of the first \( n \) terms of the sequence:
\[ S_n = \sum_{k=5}^{24} \frac{1}{k(k+1)} \]

### Step 4: Simplify the expression
Notice that:
\[ \frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1} \]
This means that the series is telescoping.

### Step 5: Write the sum explicitly
Thus, we can write:
\[ S = \left( \frac{1}{5} - \frac{1}{6} \right) + \left( \frac{1}{6} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{8} \right) + \ldots + \left( \frac{1}{24} - \frac{1}{25} \right) \]

### Step 6: Evaluate the telescoping series
When we sum these terms, we see that all intermediate terms cancel out:
\[ S = \frac{1}{5} - \frac{1}{25} \]

### Step 7: Calculate the final result
Now, we can compute:
\[ S = \frac{1}{5} - \frac{1}{25} = \frac{5}{25} - \frac{1}{25} = \frac{4}{25} \]

### Final Answer
Thus, the sum of the first 20 terms of the sequence is:
\[ \boxed{\frac{4}{25}} \]