Find the sum of:
$(i)$ the first $1000$ positive integers
$(ii)$ the first $n$ positive integers

(N/A) $(i)$ Let $S = 1 + 2 + 3 + \ldots + 1000$.
Using the formula $S_n = \frac{n}{2}(a + l)$ for the sum of the first $n$ terms of an Arithmetic Progression $(AP)$,where $a = 1$ and $l = 1000$:
$S_{1000} = \frac{1000}{2}(1 + 1000) = 500 \times 1001 = 500500$.
So,the sum of the first $1000$ positive integers is $500500$.
$(ii)$ Let $S_n = 1 + 2 + 3 + \ldots + n$.
Here,the first term $a = 1$ and the last term $l = n$.
Therefore,$S_n = \frac{n(1 + n)}{2}$ or $S_n = \frac{n(n + 1)}{2}$.
So,the sum of the first $n$ positive integers is given by $S_n = \frac{n(n + 1)}{2}$.