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Q.
The largest term of the sequence $\frac{1}{503}, \frac{4}{524}, \frac{9}{581}, \frac{16}{692}, \ldots$ is
Sequences and Series
Solution:
The general term of the given sequence is
$T_{n}=\frac{n^{2}}{500+3 n^{3}}$
then, $\frac{d T _{n}}{d n}=\frac{n\left(1000-3 n^{3}\right)}{\left(500+3 n^{3}\right)^{2}}$
For maximum or minimum of $T_{n}$
$\frac{d T _{n}}{d n}=0$
$\therefore n=\left(\frac{1000}{3}\right)^{\frac{1}{3}}$
Now, $6<\left(\frac{1000}{3}\right)^{1 / 3}< 7$
Hence, $T_{7}$ is largest term.
So largest term in the given sequence is $\frac{49}{1529}$.