<s>[INST] Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$ Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
 [/INST] Let $k$ be any number. We know that $(k+1)² - k² = 2k + 1$
Let $k = 1, 2, 3, 4, ..., n$ 
$$(1+1)² - (1)²= 2(1) + 1$$
$$(2+1)² - (2)²= 2(2) + 1$$
$$(3+1)² - (3)²= 2(3) + 1$$
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$$(n+1)² - (n)²= 2(n) + 1$$
Adding all the equations, we get: 
$$- (1)² + (n+1)² = 2(1 + 2 + 3 + ... + n) + n(1)$$
$$n² + 2n = 2(1 + 2 + 3 + ... + n) + n$$
$$2(1 + 2 + 3 + ... + n) = n² + n$$
$$\therefore (1 + 2 + 3 + ... + n) = n(n+1)/2$$
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