I was studying the Circle Division Problem, which is perfectly described with a solution here: LINK
But for now I have quite similar problem, the difference is the cutting line:
So, I was trying to manage to find recurrent formula for this specific problem, but failed. Could anyone give me the hint in what way I should start thinking?
What I managed to find: 1 slice - 2 pieces, 2 slices - 12 pieces, 3 slices - 31 pieces (?)
And when I cut 2nd time so I hit the first line totally 9 times, might this be helpful?
Thank you all for your time and suggestions, I'm still quite a beginner in this field.
Okay, so somehow I managed to first find the recurrent formula, which is: $$ f(n) = f(n-1) + 9(n-1) +1$$ Then, based on this formula, I've tried to make sequence: $$ f(n) = f(n-1) +9(n-1)+1=f(n-2)+9(n-2)+9(n-1)+1+1=f(n-3)+9(n-3)+9(n-2)+9(n-1)+1+1+1 = ...=f(1)+1+1+...+1+9((n-1)+(n-2)+...+1)= f(1) + n-1+9(n(n-1)/2) = 2+n-1+(9n^2-9n)/2 = (2+2n+9n^2-9n)/2 = (9n^2-7n+2)/2 $$
Also I've tried to prove this formula with induction and it worked well, just include above mentioned recurrent formula while proving.