The number 2025 is a very interesting number, which makes the year 2025 a very interesting year ...
The number 2025 = 452 is a perfect square. The last time we had a perfect square year was the year 1936 = 442, which was 89 years ago. And the next time we'll have a perfect square year will be the year 2116 = 462, which is 91 years into the future. So for most of us, the year 2025 is the only perfect square year we experience in our whole life.
Also, perfect squares have this property:
k2 = 1 + 3 + 5 + ... + (2k − 1)
So, we can also write 2025 as a sum of the first 89 odd integers:
2025 = 452 = 1 + 3 + 5 + ... + 89.
The number 45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 is a triangular number. So we have
2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)2.
The last time we had a squared triangular number year was the year 1296 = ((1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)2, and the next time we'll have a squared triangular number year will be the year 3025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)2. So we are really lucky to experience that in our short life.
The square of a triangular number has this very special property:
(1 + 2 + 3 + ... + n)2 = 13 + 23 + 33 + ... + n3
So we have:
2025 = (1 + 2 + 3 + ... + 9)2 = 13 + 23 + 33 + ... + 93,
which is the sum of the first nine cubes!
