If {t_n} = frac{1}{4}(n + 2)(n + 3) for n = 1 | vedclass

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(D) Given ${t_n} = \frac{1}{4}(n + 2)(n + 3)$.We need to find the sum $S = \sum_{n=1}^{2003} \frac{1}{t_n} = \sum_{n=1}^{2003} \frac{4}{(n+2)(n+3)}$.U

Vedclass · Accepted Answer

$\frac{4006}{3009}$

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(D) Given ${t_n} = \frac{1}{4}(n + 2)(n + 3)$.We need to find the sum $S = \sum_{n=1}^{2003} \frac{1}{t_n} = \sum_{n=1}^{2003} \frac{4}{(n+2)(n+3)}$.Using the method of partial fractions,$\frac{1}{(n+2)(n+3)} = \frac{1}{n+2} - \frac{1}{n+3}$.Thus,$S = 4 \sum_{n=1}^{2003} \left( \frac{1}{n+2} - \frac{1}{n+3} ight)$.This is a telescoping series:$S = 4 \left[ \left( \frac{1}{3} - \frac{1}{4} ight) + \left( \frac{1}{4} - \frac{1}{5} ight) + \dots + \left( \frac{1}{2005} - \frac{1}{2006} ight) ight]$.$S = 4 \left( \frac{1}{3} - \frac{1}{2006} ight) = 4 \left( \frac{2006 - 3}{3 	imes 2006} ight) = 4 	imes \frac{2003}{6018} = \frac{8012}{6018} = \frac{4006}{3009}$.

If ${t_n} = \frac{1}{4}(n + 2)(n + 3)$ for $n = 1, 2, 3, \dots$,then $\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + \dots + \frac{1}{t_{2003}} = $

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