# Math Help - Sum of odd integers

1. ## Sum of odd integers

How to prove that $1+3+5+...+n = \frac{(n+1)^2}{4}$ ?

2. In a word: induction.

In a thousand words:
Code:
1 3 5 7 9
3 3 5 7 9
5 5 5 7 9
7 7 7 7 9
9 9 9 9 9

3. Hello, OReilly!

Here's one way . . .

How to prove that: $1+3+5+...+n = \frac{(n+1)^2}{4}$ ?
This is an arithmetic series
. . with first term $a = 1$, common difference $d = 2$, and $\frac{n+1}{2}$ terms

Using the Sum formula: . $S \:= \:\frac{1}{2}\cdot\frac{n+1}{2}\left[2\cdot1 + 2\cdot\left(\frac{n+1}{2} - 1\right)\right]$

. . . $= \;\frac{n+1}{4}[2 + (n + 1 - 2)] \;= \;\frac{n+1}{4}(n+1) \;= \;\frac{(n+1)^2}{4}$