1. What is the Sum of Odd Numbers Formula?

The sum of the first n odd numbers is given by the simple formula n². This elegant mathematical relationship shows that the sum grows quadratically with the number of terms.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( n \) — Number of odd terms to sum

Example: For n=5, the first 5 odd numbers are 1, 3, 5, 7, 9. Their sum is 1+3+5+7+9 = 25, which equals 5².

3. Mathematical Proof

Proof by Induction:

1. Base case: For n=1, sum=1=1²
2. Assume true for n=k: 1+3+...+(2k-1)=k²
3. For n=k+1: sum = k² + (2(k+1)-1) = k² + 2k + 1 = (k+1)²

Thus, by induction, the formula holds for all positive integers n.

4. Practical Examples

Example 1: n=4 → Sum=16 (1+3+5+7=16)
Example 2: n=10 → Sum=100 (1+3+...+19=100)
Example 3: n=20 → Sum=400 (1+3+...+39=400)

5. Frequently Asked Questions (FAQ)

Q1: Why does the sum of odd numbers equal n²?
A: This is a fundamental mathematical identity that can be proven by induction or visualized geometrically as forming perfect squares.

Q2: What's the largest n this calculator can handle?
A: Technically limited only by PHP's integer size, but practically any reasonable number you'd need.

Q3: Does this work for fractional or negative n?
A: No, n must be a positive integer since you can't have a fractional number of terms.

Q4: How is this formula useful in real life?
A: It's used in number theory, algorithm analysis, and mathematical proofs. It also helps understand quadratic growth patterns.

Q5: What's the connection to triangular numbers?
A: While sum of first n odds is n², sum of first n numbers is n(n+1)/2 (triangular numbers). Both show interesting patterns in number theory.