Which of the following statement(s) is/are TRUE? \({\rm{I}}.\;\left( {1{\rm{\;}} + {\rm{\;}}\frac{1}{2}} \right)\left( {1{\rm{\;}} + {\rm{\;}}\frac{1}{3}} \right)\left( {1{\rm{\;}} + {\rm{\;}}\frac{1}{4}} \right) \ldots \left( {1{\rm{\;}} + {\rm{\;}}\frac{1}{{998}}} \right) > 497\) \({\rm{II}}.\;14\frac{3}{4}{\rm{\;}} + {\rm{\;}}5\frac{1}{4} - 2\frac{1}{2} > 11\frac{1}{8}{\rm{\;}} + {\rm{\;}}12\frac{3}{8} - 7\frac{1}{4}\)
Both I and II
Statement I: Each factor \(1+\tfrac{1}{n}=\tfrac{n+1}{n}\). The product telescopes:
\[\frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdots \frac{999}{998} = \frac{999}{2} = 499.5 > 497\] True.
Statement II: LHS = \(14\tfrac{3}{4}+5\tfrac{1}{4}-2\tfrac{1}{2} = 17\tfrac{1}{2}\); RHS = \(11\tfrac{1}{8}+12\tfrac{3}{8}-7\tfrac{1}{4}=16\tfrac{1}{4}\). Since \(17.5 > 16.25\), True.
Hence Both I and II.
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