Finding Person 1's values whilst only knowing Person 7's using iteration The general formula is $a_{n+1}=a_n-\sqrt{3}b_n$ and $b_{n+1}=\sqrt{3}a_n+b_n$.
Person 7's numbers are $a_7=8$ and $b_7=32$.
Firstly, I tried to solve it without iteration and just kept changing my values until I reached Person 1's numbers, but then I realised that I had probably got it wrong, and it was taking way too long. So afterwards, I attempted to create a general equation that had all values for $a$ and $b$ from 1 to 7. I'm not sure if that was a useless step, as I am unable to go anywhere with this information.
$$32-8\sqrt{3}=\sqrt{3}(a_1+a_2+a_3+a_4+a_5+a_6)+3(b_2+b_3+b_4+ b_5+ b_6)+4b_1$$
The above was my final step before I was at a loss and decided to come here for help.

(This process works because of how unique the numbers are)
Consider the complex number $ c_n = a_n + i b_n$.
Find a relationship between $ c_n$ and $ c_{n+1}$.

 $ c_{n+1} = c_n \times 2 e^{ \frac{i \pi } { 3 } }$.

Hence, find a relationship between $c_1$ and $c_7$.

 $c_7 = 2^6 c_1$.

Hence, find $ a_1, b_1$.
