The problem is to find the value of,
$\sqrt{-1+1\sqrt{-2+2\sqrt{-3+3\sqrt{-4+4\sqrt{...}}}}}$
Even though I have solved the problems which have a definite pattern which repeats itself and we make some substitution to get a quadratic which gives the final answer but I can't think of a way to follow that over here. Can anyone share the approach for this problem?
Note: I just came up with this problem (maybe it already exists) but then I wanted to know how could one solve such problems seeing it for the first time. I have verified it for some values and I believe the problem statement is correct. The method of generating this expression has been provided in the comments.
OK. My reaction based on your explanation in the comments.
Do not say $$ 1 = \sqrt{-1+1\sqrt{-2+2\sqrt{-3+3\sqrt{-4+4\sqrt{\dots}}}}} \tag1$$ Instead say: one solution of the recurrence $$ a_n^2 = -n+na_{n+1} \tag2$$ is $a_n =n$. The reason for saying it this way is that any other solution of the recurrence has just as much claim to be called the "value" of the infinite nested radical as this solution has.
Note that given any number $r$, there is a solution $(a_n)$ of the recurrence $(2)$ with $a_1 = r$.
Another, more conventional, interpretation of the problem would ask for the limit (if it exists) for the sequence \begin{align} a_1 &= \sqrt{-1} = 0.0000000000 + 1.0000000000 i \\ a_2 &=\sqrt{-1+1\sqrt{-2}} =0.6050003336 + 1.168770894 i \\ a_3 &= \sqrt{-1+1\sqrt{-2+2\sqrt{-3}}} =0.9306048592 + 0.9306048592 i \\ &\qquad\vdots \end{align} Always taking the principal square root. Numerically, it seems this converges to $1$. Can this be proved?