Compute the sum of the integers from 1 to 1000.

$S = 1 + 2 ~ + ~ ... ~ + ~ 500 + 501 ~ + ~ ... ~ + ~ 999 + 1000$

$S = (1 + 1000) + (2 + 999) + ~ .... ~ + ~ (500 + 501)$

$S = 1001 + 1001 + ~ .... ~ + ~ 1001$

$S = 1001 \times 500$

$S = 500,500$

$S = 1 + 2 + 3 ~ + ~ ... ~ + ~ 1000$   ←   Sum of Arithmetic Progression

$S = \dfrac{n}{2}(a_1 + a_n)$

$S = \dfrac{1000}{2}(1 + 1000)$

$S = 500,500$