Odd Number Theorem/Visual Demonstration

Theorem

$\ds \sum_{j \mathop = 1}^n \paren {2 j - 1} = n^2$

That is, the sum of the first $n$ odd numbers is the $n$th square number.

Visual Demonstration

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $25$
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.2$: Pythagoras (ca. $\text {580}$ – $\text {500}$ B.C.)
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $25$