It’s visually easy to see that the sum of consecutive odd numbers forms perfect squares—this brilliant animation is perfect for empirically understanding why. But how can we explain it in words?
1️⃣ The sum of consecutive odd numbers produces perfect squares
– The sequence of odd numbers:
1, 3, 5, 7, 9, …
– The sum of the first n odd numbers follows the formula:
1 + 3 + 5 + … + (2n-1) = n²
– This can be proven by induction.
2️⃣ The discrete derivative of n² is 2n + 1
– The discrete derivative (forward difference) of a function f(n) is:
Δ f(n) = f(n+1) – f(n)
– Applying it to f(n) = n²:
(n+1)² – n² = n² + 2n + 1 – n² = 2n + 1
– This shows that the difference between consecutive squares is always an odd number—specifically, the (n+1)th odd number!
A simple—well, for those who love math—yet beautiful mathematical insight!
