Number of Cuts in the circle

Question

We have a cake which we have to cut vertically into many pieces not necessarily equal,
but we have to follow some rules.

$> $Each cut has to pass through all the previous cuts.
$> $No three cuts intersect at a single point.
After the $100$th cut,how many pieces will we have in all?
$A)5051$
$B)5050$
$C)5053$
$D)5052$

 

Comments

please anyone ,solve this question, please?

Answer 1

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

Comments

I was also solving in this manner but I was not able to form a recurrence relation of the no. of regions. In recurrence relation after applying back substitution, how do you get the +1 term ? 

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due to T(2) = 4