Factorials in a nested radical?

Algebra Level 5

$\large \sqrt { 2!^{ 2 }+2!\sqrt {3!^2+3!\sqrt{4!^2+4!\sqrt{\cdots } }}}= \, ?$

1 solution

Hamza A
Feb 13, 2016

observe that $n! + (n + 1)! =\sqrt{n!^2 + n! [(n + 1)! + (n + 2)!]}$

expanding we get

$n! + (n + 1)!=\sqrt{n!^2 + n!\sqrt{(n + 1)!^2 + (n + 1)!\sqrt{(n + 2)!^2 + . . .}}}$

this is exactly what the problem is asking,plugging in $n=2$ we have

$a=2!+3!=8$