Question

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

• A. $\frac{16}{692}$
• B. $\frac{4}{524}$
• C. $\frac{49}{1529}$
• D. None of these

Answer 1

Answer: (C)

The general term of the given sequence is
$T_n=\frac{n^2}{500+3n^3}$
then, $\frac{d{T}_n}{dn}=\frac{n\left(1000-3n^3\right)}{{\left(500+3n^3\right)}^2}$
For maximum or minimum of $T_n$
$\frac{d{T}_n}{dn}=0$
$\therefore n={\left(\frac{1000}{3}\right)}^{\frac{1}{3}}$
Now, $6<{\left(\frac{1000}{3}\right)}^{1/3}<7$
Hence, $T_7$ is largest term.
So largest term in the given sequence is $\frac{49}{1529}$.