Sum of Odd Numbers — Formula, Proof & Examples
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.