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For $n\geq 4$, $1!+2!+3!+4!+.....n!$ is always

$\begin{array}{1 1} Even \;number \\ Square\; of\; a\; positive number \\ Not\; a\; square\; of\; any\; positive\; integer \\ Odd\; and\; square\; of\; a\; positive\; integer \end{array}$

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Since $n\geq 4$ $n=5\:or\:6\:or\:7.....$
$1!+2!+3!+4!=33$
5!,6!,..............n! all these numbers end with 0.
$\therefore$$1!+2!+3!+4!+5!+..........n! = 33$ + (any number ending with 0)
= any number ending with 3.
$\Rightarrow\: $ It cannot be even, cannot be a square number.
$\therefore$ It is not a square of any positive integer.
answered Sep 6, 2013 by rvidyagovindarajan_1
 

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