Question

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

Answer 1

Answer:

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

Step-by-step explanation:

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

S₁ = 1 + 2 + 3 + 4 + ......... + 99

⇒ S₁ = $\frac{99 \times (99 + 1)}{2} = 4950$

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 = $3 \times \frac{33 \times (33 + 1)}{2} = 1683$

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 = $\frac{n \times (n + 1)}{2}$ .