Happy new year 2025

It’s interesting to note that \(2025 = 1^3 + 2^3 + ... + 9^3\)

( It is also equal to \((1 + 2 + ... + 9)^2\) )

You can verify it quickly with declaring R functions on the prompt:

f <- function(k) {
  sum((1:k)^3)
}

g <- function(k) {
  sum(1:k)^2
}

paste("f(9) =", f(9)); paste("g(9) =", g(9))

[1] "f(9) = 2025"
[1] "g(9) = 2025"

The last time the year was expressed as one of these series was in \(f(8) =1296\); next time will be in \(f(10) =3025\) (which is \(f(9)+10^3\), i.e. \(2025 + 1000\)).

Interestingly, this generalizes to all integers, the following expression is true for any N:

\[(1^3 + 2^3 + ... + N^3) = (1 + 2 + ... + N)^2\]
or, in more formal mathematical notation:

\[ \sum_{i=1}^Ni^3 = \left(\sum_{i=1}^Ni\right)^2 \]

To prove it, I used a proof by induction (and the fact that \(1+2+..+N=\frac{N(N+1)}{2}\)).

Can you?