HMMT 2011 T5

Let $a$ and $b$ be positive real numbers. Define two sequences of real numbers $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ for all positive integers $n$ by $(a+b i)^{n}=a_{n}+b_{n} i$. Prove that

$$\frac{\left|a_{n+1}\right|+\left|b_{n+1}\right|}{\left|a_{n}\right|+\left|b_{n}\right|} \geq \frac{a^{2}+b^{2}}{a+b}$$

for all positive integers $n$.