I have the below difference equation, which I need to find a general solution for:
$a_{n+1} = a_n^2 − 10,\ a_0 = 4$
I've done simpler problems which I know how to reach a general solution for. However, I'm unsure of how to handle this particular problem.
I've seen in some textbooks that they deem certain equations to be too complex.
Is this realistically solvable and if so what steps can I take to reach a solution?
You could do something by yourself since, if the was no constant term, taking logarithms, it is simple to show that $$a_{n+1}=a_n^2 \implies a_n=e^{c\, 2^n}$$
For your sequence, the first terms being $$\{4,6,26,666,443546,196733054106\}$$ use as initial condition $a_6=196733054106$ and this will give $$e^{64 c}=196733054106\implies c=\frac 1 {64} \log(196733054106)$$ and tjis will give as an approximation
$$a_n=[196733054106]^{2^{n-6}}$$
This would give $$a_7 \sim 38703894577874323459236$$ instead of the exact $$a_7=38703894577874323459226$$
$$a_8=1497991455495209454914700643437714405353703696$$ instead of the exact $$a_8=1497991455495209454913926565546156918884519066$$
