Which of the following statement are true
I. `1/(1 times 3) +1/(3 times 5)+1/( 5 times 7)+……….+ 1/(11 times 13)=12/13`
II. `1/(1 times 2) +1/( 2 times 3)+1/( 3 times 4)+…..+1/(12 times 13)=12/13`

Options

A. Only I
B. only II
C. Both I and II
D. neither I nor II

Answer

To solve the problem, we need to evaluate the two statements given and determine their validity.

### Statement I:
**Evaluate the series:**
\[ S_1 = \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \frac{1}{5 \times 7} + \ldots + \frac{1}{11 \times 13} \]

1. **Rewrite each term:**
Each term can be expressed as:
\[ \frac{1}{n(n+2)} = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+2} \right) \]
Thus, we can rewrite \( S_1 \) as:
\[ S_1 = \frac{1}{2} \left( \left( \frac{1}{1} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{5} \right) + \left( \frac{1}{5} - \frac{1}{7} \right) + \ldots + \left( \frac{1}{11} - \frac{1}{13} \right) \right) \]

2. **Observe cancellation:**
Notice that in the series, all intermediate terms cancel out:
\[ S_1 = \frac{1}{2} \left( 1 - \frac{1}{13} \right) = \frac{1}{2} \left( \frac{13 - 1}{13} \right) = \frac{1}{2} \cdot \frac{12}{13} = \frac{6}{13} \]

3. **Conclusion for Statement I:**
Since \( S_1 = \frac{6}{13} \) and not \( \frac{12}{13} \), **Statement I is false.**

### Statement II:
**Evaluate the series:**
\[ S_2 = \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \ldots + \frac{1}{12 \times 13} \]

1. **Rewrite each term:**
Each term can be expressed as:
\[ \frac{1}{n(n+1)} = 1 - \frac{1}{n+1} \]
Thus, we can rewrite \( S_2 \) as:
\[ S_2 = \left( 1 - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \ldots + \left( \frac{1}{12} - \frac{1}{13} \right) \]

2. **Observe cancellation:**
Again, notice that in the series, all intermediate terms cancel out:
\[ S_2 = 1 - \frac{1}{13} = \frac{13 - 1}{13} = \frac{12}{13} \]

3. **Conclusion for Statement II:**
Since \( S_2 = \frac{12}{13} \), **Statement II is true.**

### Final Conclusion:
- **Statement I:** False
- **Statement II:** True