Question

Determine the sum of the Natural numbers up to 1000 which are not divisible by 5

Answer 1

Sum the first 1000 natural numbers:

$\displaystyle 1 + 2 + 3 + \cdots + 100 = \sum_{i=1}^{1000} i = \frac{1000 \cdot 1001}2 = 500,500$

Since 1000/5 = 200, there are 200 multiples of 5 in the range 1-1000. Sum them up:

$\displaystyle 5 + 10 + 15 + \cdots + 1000 = 5 (1 + 2 + 3 + \cdots + 200) = 5\sum_{i=1}^{200}i = 5\cdot\frac{200\cdot201}2 = 100,500$

Then the sum we want is

$\displaystyle 1 + 2 + 3 + 4 + 6 + 7 + 8 + 9 + 11 + \cdots + 999 = 500,500-100,500 = \boxed{400,000}$