Convergence of the sequence $x_n = \sqrt{x_{n-1}x_{n-2}}$ for all $n\geq3$

Question

Let $\left\{x_n\right\}$ be a sequence of positive real numbers such that $x_n = \sqrt{x_{n-1}x_{n-2}}$ for all $n\geq3$. Then show that the sequence converges to $(x_1x_2^2)^{1/3}$.

How can I do the sum? I have learnt Squeeze theorem, ratio test and root test for convergence of sequence, but I am not sure whether those theorems will help in solving this sum. Please give some hint to approach it.

Please anyone help me solve it. Thanks in advance.

Answer 1

Note that $(x_{n+1})^2 x_n = x_n x_{n-1} x_n = x_n^2 x_{n-1}$, and so this is constant. So if $x_n$ converges to $a$ (we know it converges by the monotone convergence theorem), we have $a^3 = x_1 x_2^2$, as required.