imported>Unknown at 05:23, 7 February 2021

2021-02-07T05:23:38Z

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==== Recurrent Problems - Lines In The Plane ====
What is the maximum number Ln 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''', '''Ln = Ln - 1 + n, for n > 0'''

===== closed form =====
Ln = Ln - 1 + n
= Ln - 2 + (n - 1) + n
= Ln - 3 + (n - 2) + (n - 1) + n
. . . .
= L0 + 1 + 2 + ... + (n - 2) + (n - 1) + n
= 1 + Sn, where Sn = 1+ 2 + 3 + ... + (n - 2) + (n - 1) + n.

Sn = 1+ 2 + 3 + ... + (n - 2) + (n - 1) + n
+Sn = n + (n - 1) + (n - 2) + ... + 3 + 2 + 1
----
2Sn = (n + 1) + (n + 1) + (n + 1) + ... + (n + 1) + (n + 1) + (n + 1)

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

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

Lines In The Plane - Revision history

imported>Unknown at 05:23, 7 February 2021