How the formula for mean energy of quantum harmonic oscillator is derived?

Remember the partition function is $$Z = sum_E e^{-beta E}$$ But $E=hbar omega (n+frac12)$ so $$Z = sum_{n=0}^infty (e^{-beta hbar omegacdot frac12 }cdot e^{-beta hbar omegacdot n })=e^{-beta frac{hbar omega}{2}}sum_{n=0}^infty (e^{-beta hbar omega })^n = e^{-beta frac{hbar omega}{2}}cdot frac{1}{1-e^{-beta hbar omega}}$$ (The sum is determined by using the sum of an infinite geometric series)

Now just use $langle Erangle = sum Ecdot e^{-beta E} /Z$ (again use wisely the sum of an infinite series) or any other formula from statistical mechanics (deriving the partition function by beta and taking a Log...) and you'll get your answer