# Sum of Odd Numbers — Formula, Proof & Examples

TL;DR

The sum of odd numbers follows one clean rule: the first n odd numbers add up to n². This article gives the formula 1+3+5+⋯+(2n−1)=n², a visual square-building proof, six worked examples, and the mistakes that trip students up.

The **sum of the first n odd numbers is n²**. Written as a formula: 1+3+5+⋯+(2n−1)=n² Here (2n−1) is the n-th odd number: the 1st is 2(1)−1=1, the 4th is 2(4)−1=7, and so on. An **odd number** is any integer not divisible by 2, so the odd numbers are 1,3,5,7,9,…, each two more than the one before.

Because each term is 2 more than the last, the odd numbers form an arithmetic progression with first term 1 and common difference 2 — which is why a tidy closed formula exists at all.

## Sum of Odd Numbers Formula

The sum of the first n odd numbers is: 1+3+5+⋯+(2n−1)=n²

| Symbol         | Meaning                                                 |
|----------------|---------------------------------------------------------|
| n              | The count of odd numbers being added (not the last term) |
| 2n−1          | The n-th odd number — the last term in the run        |
| n²            | The total, always a perfect square                      |

The formula can also be reached through the arithmetic-progression sum S_n = n/2(a+l), with first term a=1 and last term l=2n−1: S_n = n/2(1+(2n−1))=n².

## Examples of the Sum of Odd Numbers

### Example 1

**Find the sum of the first 5 odd numbers.**

The first 5 odd numbers are 1,3,5,7,9. Here n=5. Sum=n²=5²=25. Check by adding: 1+3+5+7+9=25. ✓

### Example 2

**Find the sum of the first 10 odd numbers.**

First find the 10th odd number: 2(10)−1=19. Sum= (1 + 19)/2 × 10 = 10 × 10 = 100. The direct rule is faster and shows _why_. Sum=n²=10²=100.

### Example 3

**What is the sum 1+3+5+⋯+99?**

The last term is 99, so solve 2n−1=99. 2n=100, n=50. Sum=50²=2500.

### Example 4

**The sum of the first n odd numbers is 64. Find n.**

n²=64, n=√64=8. **There are 8 odd numbers**, namely 1,3,5,7,9,11,13,15.

### Example 5

**Find the sum of odd numbers from 1 to 15.**

The odd numbers from 1 to 15 are 1,3,5,7,9,11,13,15, that is 8 terms. Sum=8²=64.

### Example 6

**Find the sum of the first 20 odd numbers, then subtract the sum of the first 12.**

Sum of first 20: 20²=400. Sum of first 12: 12²=144. Difference: 400−144=256. **The sum of the 13th through 20th odd numbers is 256.**

## Why the Sum of Odd Numbers Builds a Square

The formula is not a lucky accident — it is a picture. Start with one dot, a 1×1 square. To grow it into a 2×2 square you add an L-shaped layer along one side and the bottom: that L holds exactly 3 dots. Each layer completes the next perfect square.

So 1+3+5+⋯+(2n−1) _is_ the n×n square, dot for dot.

## Properties of the Sum of Odd Numbers

A few properties follow directly from the n² rule:

- **The sum is always a perfect square.**  
- **The odd numbers form an arithmetic progression.** First term 1, common difference 2, n-th term 2n−1.  
- **The mean of the first n odd numbers is n.**  
- **Runs that do not start at 1 need adjusting.** 
    
## Tripping Points to Avoid When Calculating the Sum of Odd Numbers

### Mistake 1: Using n² when the count is wrong

### Mistake 2: Including even numbers by accident

### Mistake 3: Assuming the rule works for odd numbers not starting at 1

## Conclusion

- The **sum of odd numbers** starting at 1 is always a perfect square: 1+3+⋯+(2n−1)=n².
- The n-th odd number is 2n−1, so convert a last-term to a term-count before squaring.
- The visual proof builds an n×n square from L-shaped layers of 1,3,5,… dots.
- The rule needs the run to start at 1; otherwise subtract the missing head of the sequence.