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Question:

Consider a sequence where the n^th term, $t_{n} = \frac{n}{n + 2}, n = 1, 2, . . . .$

The value of $t_{3} \times t_{4} \times t_{5} \times . . . \times t_{53}$ equals:

Explanation:

$t_{n} = \frac{n}{n + 2}$

$∴ t_{3} = \frac{3}{3 + 2} = \frac{3}{5}$

$t_{4} = \frac{4}{4 + 2} = \frac{4}{6}$

$t_{5} = \frac{5}{5 + 2} = \frac{5}{7}$

...

$t_{51} = \frac{51}{53}$

$t_{52} = \frac{52}{54}$

$t_{53} = \frac{53}{55}$

∴ t₃ × t₄ × t₅ × ... × t₅₃ = $\frac{3}{5}$ × $\frac{4}{6}$ × $\frac{5}{7}$ × ... × $\frac{51}{53}$ × $\frac{52}{54}$ × $\frac{53}{55}$

$= \frac{(3 \times 4)}{(54 \times 55)} = \frac{2}{495}$

Hence, option (a).

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