Why is the maximum number of electrons in a shell given by 2n²?

Each shell (principal quantum number n) contains n subshells, and the number of orbitals in a shell is n². For example, n = 2 has one s orbital and three p orbitals, giving 4 = 2² orbitals. Now, each orbital can hold a maximum of 2 electrons (because of the two possible spin values, plus and minus half, by Pauli's exclusion principle). So the maximum electrons = 2 times the number of orbitals = 2 times n² = 2n². That is why K (n=1) holds 2, L (n=2) holds 8, M (n=3) holds 18, and N (n=4) holds 32. So 2n² is not a random formula, it falls directly out of counting orbitals and doubling for spin.