No three cuts intersect at a single point.
this means no line is intersect to any other line.
take a circle(cake), ===> No.of regions (peices) = 1
First cut, will touch 1 regions ===> the 1 region is double ===> for the previous total regions, 1 region add as extra
∴ Total regions after 1st cut = 2
Second cut, will touch 2 regions ===> the 2 regions is double ===> for the previous total regions, 2 region add as extra
∴ Total regions after 2nd cut = 2+2 = 4
Third cut, will touch 3 regions ( Draw the line and observe ) ===> the 3 regions is double ===> for the previous total regions, 3 region add as extra
∴ Total regions after 3rd cut = 4+3 = 7
Fourth cut, will touch 4 regions ( Draw the line and observe ) ===> the 4 regions is double ===> for the previous total regions, 4 region add as extra
∴ Total regions after 4th cut = 7+4 = 11
T(n) = T(n-1) + n ====> if it is nth cut, then it will add n regions to T(n-1) regions
take T(2) = 4 as base condition.
By the Back-substitution method, T(n) = $\frac{n(n+1)}{2}$ + 1
∴ substitute n=100 ===> 50*101 + 1 = 5050+1 = 5051