Which of the following statement(s) is/are TRUE? \({\rm{I}}.\frac{1}{{1 \times 3}} + \frac{1}{{3 \times 5}} + \frac{1}{{5 \times 7}} + \ldots + \frac{1}{{11 \times 13}} = \frac{{12}}{{13}}\)
Only II
Use the telescoping identities \(\tfrac{1}{(2n-1)(2n+1)}=\tfrac{1}{2}\bigl(\tfrac{1}{2n-1}-\tfrac{1}{2n+1}\bigr)\) and \(\tfrac{1}{n(n+1)}=\tfrac{1}{n}-\tfrac{1}{n+1}\).
Statement I: \(\tfrac{1}{2}\bigl(1-\tfrac{1}{13}\bigr)=\tfrac{6}{13}\neq\tfrac{12}{13}\) — FALSE.
Statement II: \(1-\tfrac{1}{13}=\tfrac{12}{13}\) — TRUE.
Hence only II.
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