sum of odd numbers∗
pahio†
2013-03-21 17:57:37
The sum of the first n positive odd integers can be calculated by using the
well-known property of the arithmetic progression, that the sum of its terms is
equal to the arithmetic mean of the first and the last term, multiplied by the
number of the terms:
1 + 3 + 5 + 7 + 9 + · · · + (2n−1) = n ·
{z
}
|
1+(2n−1)
= n2
2
n
Thus, the sum of the first n odd numbers is n2 (this result has been proved first
time in 1575 by Francesco Maurolico).
Below, the odd numbers have been set to form a triangle, each nth row
containing the next n consecutive odd numbers. The arithmetic mean on the
row is n2 and the sum of its numbers is n · n2 = n3 .
1
3
7
13
21
31
..
.
11
15
23
33
5
9
17
25
35
37
..
.
19
27
29
39
41
..
.
∗ hSumOfOddNumbersi
created: h2013-03-21i by: hpahioi version: h36230i Privacy setting: h1i hExamplei h00A05i h11B25i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
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