sum of odd numbers

The sum of the first $n$ positive odd integers can be calculated by using the well-known of the arithmetic progression, that the sum of its is equal to the arithmetic mean of the first and the last , multiplied by the number of the :

$$\underset{n}{\underbrace{1+3+5+7+9+\mathrm{\cdots }+(2n-1)}}=n\cdot \frac{1+(2n-1)}{2}=n^2$$

Thus, the sum of the first $n$ odd numbers is $n^2$ (this result has been proved first time in 1575 by Francesco Maurolico).

Below, the odd numbers have been set to form a triangle, each $n^{\mathrm{th}}$ row containing the next $n$ consecutive odd numbers.  The arithmetic mean on the row is $n^2$ and the sum of its numbers is  $n\cdot n^2=n^3$.

Title	sum of odd numbers
Canonical name	SumOfOddNumbers
Date of creation	2013-03-22 14:38:35
Last modified on	2013-03-22 14:38:35
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Example
Classification	msc 00A05
Classification	msc 11B25
Related topic	NumberOdd