What is the justification for $n+\ell$ rule in quantum mechanical model of atom? So my textbook says 

Mathematically, the dependence of energies of the orbitals on $n$ and $\ell$ are quite complicated but one simple rule is that, the lower the value of $n+\ell$ for an orbital, the lower is its energy. If two orbitals have the same value of $n+\ell$, the orbital with lower value of $n$ will have the lower energy.

Here $n$ is principal quantum number and $\ell$ is azimuthal quantum number. 
What I do not get is why this $n+\ell$ rule is true? (And I am not sure if there is a derivation for this) 
The rule seems to hold up to $7p$ orbital and I cannot verify later on. 
Question:
Is this rule always true ? If yes then, why? 
 A: The reason it works is phenomenological. There is a reason based on the quantum mechanics which suggests it might work, but its actual success depends on parameters inherent in the particles involved and not in the mathematics.
The quantum mechanics of atoms gives rise to the two quantum numbers mentioned, $n$ and $\ell$, the principal and orbital/azimuthal quantum numbers. Without spin-orbit coupling, the energy would be determined by $n$ only. And a higher $n$ would have a higher energy.
With $\ell > 0$, the orbital angular momentum couples with the spin of the electron to create a splitting of the energy due to the interaction of the electron's magnetic dipole moment with the nuclear charge. Each $\ell$ state is split into two states. (Actually, there is even more splitting possible, but I won't detail that.) The amount of splitting depends on the interaction strength of those fields.
In some cases, that causes a state with a smaller $n$ to be increased in energy above a state with a lower $n$. The first case this happens is the $3d$ states will split up above the $4s$ states. Again, the amount of splitting depends on the value of the electron magnetic dipole.  If the dipole was smaller, the amount of splitting would be less and $3d$ might not be above $4s$, but then the world would be quite different.
I don't know if it's always true but in heavier atoms (and larger $n$), relativistic effects accentuate the spin-orbit splitting.
But basically, it's a handy tool or trick which generally works.
