Q32 / 90 · KVPY 2017 (19 Nov SB/SX) · Mathematics
Among the inequalities below, which ones are true for all natural numbers n greater than 1000 ?
I. $\mathrm{n} ! \leq \mathrm{n}^{\mathrm{n}}$
II. $(n !)^{2} \leq n^{n}$
III. $10^{\mathrm{n}} \leq \mathrm{n} !$
IV. $n^{\mathrm{n}} \leq(2 \mathrm{n}) !$
- AI and IV only
- Correct answerBI, III and IV only
- CII and IV only
- DI, II, III and IV
Solution
(A) $\frac{\mathrm{n}^{\mathrm{n}}}{\mathrm{n} !}=\left(\frac{\mathrm{n}}{\mathrm{n}}\right)\left(\frac{\mathrm{n}}{\mathrm{n}-1}\right)\left(\frac{\mathrm{n}}{\mathrm{n}-2}\right) \ldots \ldots .\left(\frac{\mathrm{n}}{1}\right) \geq 1$
$\mathrm{n}^{\mathrm{n}} \geq \mathrm{n} !$ (correct)
(C) $\frac{\mathrm{n} !}{10^{\mathrm{n}}}=\frac{(\mathrm{n})(\mathrm{n}-1)(\mathrm{n}-2) \ldots \ldots}{(10)(10) \ldots \ldots \mathrm{n} \text { times }}$
given that $\mathrm{n}>1000$
clearly $\frac{\mathrm{n} !}{10^{\mathrm{n}}} \geq 1$
$\mathrm{n} ! \geq 10^{\mathrm{n}}$
(D)
$\frac{(1.2 .3 \cdot 4 \ldots \ldots . . n)(n+1)(n+2)(n+3) \ldots \ldots(n+n)}{n^{n}}$
$(n !)\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\left(1+\frac{3}{n}\right) \ldots \ldots .\left(1+\frac{n}{n}\right)$
clearly $\geq 1$.
$2 n ! \geq n^{n}$
Hence, the correct answer is option: 'B'.