Product Becomes Sum

$\frac { n(n+1) }{ 2 }$ is the formula for triangular numbers i.e. the sum of integers from $1$ to $n$ .

$n!$ is the notation for the product of integers from $1$ to $n$ .

$n! > \frac { n(n+1) }{ 2 }$ when $n > 3$ because $4! = 24 > 10 = \frac { 4(4+1) }{ 2 }$ and adding integers greater than 1 would always result in a smaller result than if you multiplied integers greater than 1 (with the exception of 2 * 2 and 2+2).

Therefore, $n = 1, 2 or 3$

If $n = 1$ then $1! = 1$ and $\frac { 1(1+1) }{ 2 } = 1$ . Therefore, $n = 1$

If $n = 2$ then $2! = 2$ and $\frac { 2(2+1) }{ 2 } = 3$ . Therefore, $n$ is not $2$

If $n = 3$ then $3! = 6$ and $\frac { 3(3+1) }{ 2 } = 6$ . Therefore, $n = 3$

Therefore $n = 1$ and $3$