Inspired by Zandra Vinegar

Number Theory Level 2

True or false :

$\quad\quad$ For all positive integers $a$ , $(a^2)! \geq (a!)^2$ holds true.

Clarification : $!$ denotes the factorial notation.

Inspiration .

True False Neither true nor false

1 solution

Kay Xspre
Feb 12, 2016

We know that $\frac{a^2!}{a!} \geq 1$ . (of which the equation will be true when $a = 1$ ). For $a \geq 2$ , if we divide both sides with $(a!)^2$ , we will get that

$\frac{a!}{a!} \times \prod_{i=1}^a \frac{(a+i)}{i} \times\frac{a^2!}{(2a)!} \geq 1$

Brilliant solution sir!

Mehul Arora - 5 years, 4 months ago

Inspired by Zandra Vinegar

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