A Solution to the Summing Multiples Problem

Problem:

Find the sum of all the multiples of 3 from 1 through 300. Do you see a pattern
developing? Find the sum of the multiples of 7 from 1 through 700. Have you found the
pattern? Next suppose we let N be any whole number. Find the sum of all the multiples
of N from 1 through 100*N.

(Source: Adapted from Mathematics Teaching in the Middle School, Sep-Oct1994).

Analysis:

You can use a spreadsheet to generate multiples of each of these numbers (add the
number between consecutive cells) and then use the SUM command. The table of values
below represents the sum of the multiples of N from 1 through 100*N, for N from 2 to 9.

The sum of these numbers appears to equal 5050*N

Alternate Proof:

Gauss showed that the sum of the first 100 numbers is 5050 by recognizing a pattern in the series.

He examined 1 + 2 + 3 + 4 + 5 + ... + 96 + 97 + 98 + 99 + 100, and found that the
first and last term of the series has a sum of 101 (1+100), the second and second to
last term of the series has a sum of 101 (2 + 99), the third and third to last term of the series has a sum of 101 (3 + 98), and so on, for fifty unique pairs. Thus, the sum of the numbers 1 to 100 is the same as 50*101 = 5050.

Therefore the sum of the numbers from N to 100*N

= N*(sum of numbers from 1 to 100) = N*5050, or 5050*N