Sum of Odd Numbers Calculator

About this calculator

The sum of the first n odd numbers (1 + 3 + 5 + ... up to the nth odd number) always equals a perfect square: n². This is one of the classic results in number theory, provable both algebraically and visually — arranging dots in an expanding square (a "gnomon" pattern) shows why each new odd number added extends the square by exactly one layer. It's a favorite example for introducing mathematical induction and for connecting arithmetic sequences to geometric intuition.

Beyond its role as a teaching example, this identity underlies quicker mental-math shortcuts (recognizing that 1+3+5+7+9 = 25 = 5² without adding term by term), and the underlying pattern of odd-number sums shows up in areas like discrete math, computer science algorithm analysis (where sums of this shape estimate running time), and physics problems involving uniformly accelerating quantities, since displacement under constant acceleration over successive equal time intervals follows the same odd-number progression (Galileo's odd-number rule for falling bodies).

This calculator adds up the first n odd numbers for you and shows both the total sum and how many terms were counted, confirming the n² relationship instantly.