How can I prove, the sequence converges to the given limit?

Question

Prove that the sequence $ {f(n)}$ defined by $$0<f(1)<f(2) \\ f(n+2) = [f(n+1)\cdot f(n)]^{1/2}$$ for $n \geq 1$ converges to $$[f(1) \cdot {f(2)}^2]^{1/3}$$

Answer 1

Hint: take the logarithm of $f(n)$ and turn it into a linear difference equation.

After that there's a shortcut to solve the equation, see Finding explicit formula for recursive relation