First slide

Special Series in Sequences and Series

Question

$The largest terms of the sequence \frac{1}{503}, \frac{4}{524}, \frac{9}{581}, \frac{16}{692}, \dots is$

NA

A

$\frac{16}{692}$
B

$\frac{4}{524}$
C

$\frac{49}{1529}$
D

$\frac{64}{1529}$

Solution

$T_{n} = \frac{n^{2}}{500 + 3 n^{3}} = f (n)$
$f^{1} (n) = \frac{n (1000 - 3 n^{3})}{{(500 + 3 n^{3})}^{2}}$
$\begin{array}{l} For the maxima or min f^{1} (n) = 0 \\ n = {(\frac{1000}{3})}^{1 / 3} \\ Now 6 < {(\frac{1000}{3})}^{1 / 3} < 7 \end{array}$
$∴ T_{7} is the largest term.$
$the largest term of the sequence = \frac{49}{500 + 1029} = \frac{49}{1529}$

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