Sum of All 3-Digit Multiples of 10: A Detailed Guide

Understanding the sum of all 3-digit multiples of 10 is an interesting and useful exercise in arithmetic series. This guide provides a step-by-step approach to solving this problem, making use of basic mathematical formulas and concepts.

Introduction to the Problem

The problem involves finding the sum of all numbers between 100 and 990 that are divisible by 10. These numbers form an arithmetic series where each subsequent term increases by 10.

Identifying the Series

The smallest 3-digit multiple of 10 is 100, and the largest is 990. This means we are dealing with a finite arithmetic series with a common difference of 10.

Using the Arithmetic Series Formula

Let's denote the first term of the series as ( a 100 ), the last term as ( l 990 ), and the common difference as ( d 10 ).

Step 1: Determine the Number of Terms (n)

The formula to find the number of terms ( n ) in an arithmetic series is:

n frac{l - a}{d} 1

Substituting the values:

n frac{990 - 100}{10} 1 89 1 90

This means there are 90 terms in the series.

Step 2: Calculate the Sum of the Series (Sn)

The formula to find the sum ( S_n ) of an arithmetic series is:

S_n frac{n}{2} cdot (a l)

Substituting the values:

S_{90} frac{90}{2} cdot (100 990) 45 cdot 1090 49050

Therefore, the sum of all 3-digit multiples of 10 is 49050.

Another Approach: Pair Addition

Another interesting approach is to consider the series in pairs. Each pair of terms adds up to the same value:

100 990 1090 110 980 1090 120 970 1090 ... 530 560 1090 540 550 1090

There are 45 such pairs, and thus:

45 × 1090 49050

Conclusion

In summary, the sum of all 3-digit multiples of 10 is 49050, which can be calculated using the arithmetic series formula or by pairing the terms. This method not only provides a solution but also illustrates the beauty and simplicity of arithmetic series in mathematical problem-solving.

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