Sequence Product Telescoping

CAT 2006 Slot 1 · QA · Easy · Quantitative Aptitude

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Consider a sequence where the nn-th term is tn=nn+2t_n = \frac{n}{n+2}, for n=1,2,…n = 1, 2, \dots. Find the value of t3×t4×t5×⋯×t53t_3 \times t_4 \times t_5 \times \dots \times t_{53}.

  1. A.

    2/495

  2. B.

    2/477

  3. C.

    12/55

  4. D.

    1/1485

  5. E.

    1/2970

Answer

2/495

Explanation

The product is: (35)×(46)×(57)×⋯×(5153)×(5254)×(5355)\left(\frac{3}{5}\right) \times \left(\frac{4}{6}\right) \times \left(\frac{5}{7}\right) \times \dots \times \left(\frac{51}{53}\right) \times \left(\frac{52}{54}\right) \times \left(\frac{53}{55}\right). All numerators from 5 to 53 cancel with denominators from 5 to 53. Remaining terms: Numerator = 3×4=123 \times 4 = 12. Denominator = 54×55=297054 \times 55 = 2970. Product = 122970=2495\frac{12}{2970} = \frac{2}{495}.

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