Lines In The Plane

Recurrent Problems - Lines In The Plane

What is the maximum number L_n of regions defined by lines("unfolding" or "unwinding") in the plane?

small cases

L₀ = 1 L₁ = 2 L₂ = 4 L₃ = 7 .....

mathematical expression

L₀ = 1, L_n = L_{n - 1} + n, for n > 0

closed form

L_n = L_{n - 1} + n

   = L<sub>n - 2</sub> + (n - 1) + n
   = L<sub>n - 3</sub> + (n - 2) + (n - 1) + n
   . . . .
   = L<sub>0</sub> + 1 + 2 + ... + (n - 2) + (n - 1) + n
   = 1 + S<sub>n</sub>, where S<sub>n</sub> = 1+ 2 + 3 + ... + (n - 2) + (n - 1) + n.

  S<sub>n</sub> = 1+ 2 + 3 + ... + (n - 2) + (n - 1) + n
+S<sub>n</sub> = n + (n - 1) + (n - 2) + ... + 3 + 2 + 1

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2S<sub>n</sub> = (n + 1) + (n + 1) + (n + 1) + ... + (n + 1) + (n + 1) + (n + 1)

S<sub>n</sub> = n(n + 1)/2, for n >= 0

L_n = n(n + 1)/2 + 1, for n >= 0