Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$   $\space$ ?
Let:
$$S = 1 + 2 + \ldots + (n-1) + n.$$ Write it backwards: $$S = n + (n-1) + \ldots + 2 + 1.$$
Add the two equations term by term; each addition results in $n+1.$ So:
$$2S = (n+1) + (n+1) + \ldots + (n+1) = n(n+1).$$
Divide by 2: $$S = \frac{n(n+1)}{2}.$$