Notable sums: from Gauss to even and odd numbers

Articles /notable-sums-gauss-even-odd

5 min

A chain of formulas born from a single idea.

Two rows of paired tiles and a square of tiles evoke the search for structure in sums. — Pairing terms and observing the shape of sums: a visual representation of the idea guiding this article.

1. The story: young Gauss and the sum from 1 to 100

Carl Friedrich Gauss (1777-1855) is remembered as one of the greatest mathematicians in history. According to the famous schoolroom story, when he was still a child his teacher asked the class to add the numbers from 1 to 100. Gauss noticed that, instead of carrying out a hundred additions, he could pair the first and last terms:

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
…
50 + 51 = 101

There are 50 pairs, each worth 101, so:

50 × 101 = 5050

The episode is traditionally told in biographies of Gauss. Beyond the narrative details, the important mathematical point is the idea: pair terms equally distant from the ends, because their sum is always the same.

2. Proof of the Gauss formula

We want to calculate:

S = 1 + 2 + 3 + … + (n − 1) + n

Write the same sum in reverse order:

S = n + (n − 1) + (n − 2) + … + 2 + 1

Add the two lines term by term. Each pair equals n + 1 and there are n pairs:

2S = n(n + 1)

Dividing by 2 gives:

1 + 2 + 3 + … + n = n(n + 1) / 2

3. From Gauss to the sum of the first n even numbers

We know:

1 + 2 + 3 + … + n = n(n + 1) / 2

Multiply every term by 2. This gives the first n even numbers:

2 + 4 + 6 + … + 2n

The sum is also multiplied by 2:

2 · [n(n + 1) / 2] = n(n + 1)
2 + 4 + 6 + … + 2n = n(n + 1)

4. From evens to odds

Now start directly from the first n even numbers:

2, 4, 6, 8, …, 2n

Subtract 1 from each term:

1, 3, 5, 7, …, 2n − 1

There are n terms; subtracting 1 from each has therefore subtracted n in total from the sum. Hence:

n(n + 1) − n = n² + n − n = n²
1 + 3 + 5 + … + (2n − 1) = n²

5. Evens minus odds

Let P denote the sum of the first n even numbers and D the sum of the first n odd numbers:

P = n(n + 1)      D = n²

Then:

P − D = n(n + 1) − n² = n

The same property is visible term by term:

(2 − 1) + (4 − 3) + … + [2n − (2n − 1)] = 1 + 1 + … + 1 = n

6. Evens plus odds

Adding the two formulas gives:

P + D = n(n + 1) + n² = n(2n + 1)

There is also an immediate interpretation: putting together the first n odd and the first n even numbers produces every number from 1 to 2n:

1 + 2 + 3 + … + 2n = [2n(2n + 1)] / 2 = n(2n + 1)

7. Everything in a single chain

Relationship	Formula
Sum of the first n natural numbers	1 + 2 + … + n = n(n + 1) / 2
Sum of the first n even numbers	2 + 4 + … + 2n = n(n + 1)
Sum of the first n odd numbers	1 + 3 + … + (2n − 1) = n²
Evens − odds	P − D = n
Evens + odds	P + D = n(2n + 1)

8. The teaching idea

The most interesting point is not memorising five separate formulas, but seeing how almost everything follows from the first. The Gauss formula generates the sum of the evens by simply multiplying by 2; the odds follow from the evens by subtracting 1 from each of the n terms; and the sum and difference then follow immediately from evens and odds. Thus the formulas become natural consequences of elementary transformations, not results to learn by heart.

Historical note. The schoolroom anecdote and the essential biographical details about Gauss follow the MacTutor History of Mathematics biography, University of St Andrews.