Question

The sum of \(\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \dots + \frac{1}{9 \times 10}\) is :

• A. \(\frac{10}{9}\)
• B. \(\frac{9}{10}\)
• C. \(\frac{5}{6}\)
• D. \(\frac{6}{5}\)

Hint:

For any telescoping series of the form \(\sum_{n=1}^{N} \frac{1}{n(n+1)}\), the sum is always given by the simplified formula:
\[ S_N = 1 - \frac{1}{N+1} = \frac{N}{N+1} \]
Here, since \(N = 9\), the sum is directly \(\frac{9}{9+1} = \frac{9}{10}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question asks us to find the sum of a finite series.
The series consists of terms of the form \(\frac{1}{n(n+1)}\) from \(n = 1\) to \(n = 9\).
This is a classic telescoping series where each term can be split into partial fractions.
Key Formula or Approach:
We use the partial fraction decomposition of the general term:
\[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \]
By writing each term in this difference format, adjacent terms will cancel out.

Step 2: Detailed Explanation:

• Let us write down the partial fraction decomposition for the general term of the series:
\[ a_n = \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \]

• Now, we write out the individual terms of the series from \(n = 1\) to \(n = 9\):
For \(n = 1\): \(a_1 = 1 - \frac{1}{2}\)
For \(n = 2\): \(a_2 = \frac{1}{2} - \frac{1}{3}\)
For \(n = 3\): \(a_3 = \frac{1}{3} - \frac{1}{4}\)
\(\dots\)
For \(n = 9\): \(a_9 = \frac{1}{9} - \frac{1}{10}\)

• Now, we sum these terms up to find the total sum \(S\):
\[ S = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \dots + \left(\frac{1}{9} - \frac{1}{10}\right) \]

• Observe the cancellation of terms in this sum:
The term \(-\frac{1}{2}\) cancels with \(+\frac{1}{2}\), the term \(-\frac{1}{3}\) cancels with \(+\frac{1}{3}\), and so on.
All intermediate terms cancel out, leaving only the first and the last term:
\[ S = 1 - \frac{1}{10} \]

• Simplifying this expression gives:
\[ S = \frac{10 - 1}{10} = \frac{9}{10} \]

Step 3: Final Answer:
The sum of the given series is \(\frac{9}{10}\).