Show ((n+1)^n-1)≡0 (mod n^2 ) ∀ n ∈ N(Positive Integers)

This is slightly modified Problem 14 from Waclaw Sierpinski 250 Elementary Number Theory Problems. He used vertical divides bar. The author appealed to the binomial theorem, never having to use the truth that the binomial coefficient p choose r is divisible by p a prime number. MIT Math and computer science Stalwart S´ılvia Casacuberta Puig lucidly treats a much broader version of this result.
View the lucidly presented content by David Covert November 2020
Proof Of a Modular Arithmetic Proposition