Question

Find the sum of all positive integers less than 100, which:

are not divisible by 3.

Please tell answer and formula

Answer 1

The sum of all positive integers less than 100, which are not divisible by 3 is 3267.

Explanation:

The all positive integers less than 100 has sum, given by

S₁ = 1 + 2 + 3 + 4 + ......... + 99

⇒ S₁ =
`(99 * (99 + 1))/(2) = 4950`

Now, the sum of all positive integers less than 100 which are divisible by 3 is

S₂ = 3 + 6 + 9 + 12 + 15 + ........ + 99

⇒ S₂ = 3(1 + 2 + 3 + ........ + 33)

⇒ S₂ =
`3 * (33 * (33 + 1))/(2) = 1683`

Therefore, the sum of all positive integers less than 100, which are not divisible by 3 is = S₁ - S₂ = 4950 - 1683 = 3267. (Answer)

Note : The sum of n natural numbers S is given by

S = 1 + 2 + 3 + 4 + ....... + n =
`(n * (n + 1))/(2)` .