The Sum of Multiples of 3 or 5 Below 1000: A Comprehensive Guide

The sum of all the multiples of 3 or 5 below 1000 is a classic problem that can be solved using the principle of inclusion-exclusion. This article will walk you through the step-by-step process to find the sum, explaining each part in detail.

Arithmetic Series and Sum

Let's begin by understanding how to sum the multiples of 3 below 1000. These numbers form an arithmetic series where the first term a 3, the common difference d 3, and the last term l 999. To find the number of terms n, we use the formula:

3n 999

Solving for n, we get:

n 333

The sum of the first n terms of an arithmetic series is given by:

Sn (n/2) × (a l)

Substituting the values, we find:

S3 (333/2) × (3 999) (333/2) × 1002 333 × 501 166833

Sum of Multiples of 5 Below 1000

Similarly, the multiples of 5 below 1000 form another arithmetic series:

5, 10, 15, ..., 995

For this series, the first term a 5, the common difference d 5, and the last term l 995. The number of terms n is:

5n 995

Solving for n, we get:

n 199

The sum of the first n terms is:

S5 (199/2) × (5 995) (199/2) × 1000 199 × 500 99500

Sum of Multiples of 15 Below 1000

Since the multiples of 15 are counted twice (once in the multiples of 3 and once in the multiples of 5), we need to subtract their sum to avoid double-counting. The multiples of 15 below 1000 form yet another arithmetic series:

15, 30, 45, ..., 990

For this series, the first term a 15, the common difference d 15, and the last term l 990. The number of terms n is:

15n 990

Solving for n, we get:

n 66

The sum of the first n terms is:

S15 (66/2) × (15 990) (66/2) × 1005 66 × 502.5 33165

Inclusion-Exclusion Principle

Using the inclusion-exclusion principle, we can now find the total sum of the multiples of 3 or 5 below 1000:

S S3 S5 - S15

Substituting the values, we get:

S 166833 99500 - 33165 233168

Therefore, the sum of all the multiples of 3 or 5 below 1000 is 233168.

A Visual Recap

To better understand this, let's summarize the calculations:

369...999 → 3(123...333) 3 × (333 × 334 / 2) 166833

510...995 → 5(123...198) 5 × (199 × 200 / 2) 99500

1530...990 → 15(12...65) 15 × (66 × 67 / 2) 33165

S 166833 99500 - 33165 233168