Essay on How Many Pieces

How Many Pieces?

If a line segment were to be cut into smaller segments with n number of cuts, each individual cut splits the segment into two, and any additional cut splits a newly cut segment it splits into two, which only increases the amount of segments by one; the amount of segments (S) created would be n+1. n=1 ————————————————|—————————————————
S=2
n=2
————————————|———————|—————————————
S=3 n=3 ————|————————————|————————————|————
S=4
n=20
—————|—|—|—|—|—|—|—|—|—|—|—|—|—|—|—|—|—|—|—|—————
S=21

A line segment is a one-dimensional object, and so the rule n+1 would not necessarily apply to a object of different dimensions, such as a circle, a two-dimensional object.

Number of cuts (n) | Number of separated areas within the circle (R) | 0 | 1 | 1 | 2 | 2 | 4 | 3 | 7 | 4 | 11 | 5 | 16 |

The sequence seems to be moving in a parabolic fashion.

The sequence adds n to the f(n-1), so f(n-1)= f(n-2)+n-2 and the like,
The first term f(1)=2, so R=n+1+kn(n-1) because the third term of the equation must equal zero at f(1), so the k can then be solved for with the basic formula the n outside of the parentheses is to create a parabolic function: f(2)=R(2)=4=n+1+kn(n-1) with n=an integer ≥0
So:
4=(2)+1+k(2)(n-1)
1=k(2)(1)
so k=1/2
R=n+1+(1/2)n(n-1)

R=n+1+(1/2)n(n-1)=(1/2)n(n+1)+1

Which holds true for all values found through the recursive formula.

The arithmetic sequence

Rn-1= Rn-2+n-1
Rn can be expanded to Rn= n+(n-1)+Rn-2 which becomes, through the same process;
Because R(0)=1, there must be a 1+the sequence:
R=1+n+(n-1)+(n-2)+Rn-3

and so the sum of the arithmetic sequence is given by the formula:
Sn=(n/2)(R1+Rn)
So:
R=1+(n/2)(1+Rn)
and Rn=R1+ (n− 1)d so:(d=n) R=1+(n/2)(1+(1+n(n-1))
R=1+(n/2)(2+n(n-1))=n(1+n(n-1))=
1+(1/2)n(n+1)

So:
R=0.5n(n+1)+1
R=(0.5(n-1)n)+(n+1)

If using a three dimensional object, such as a cuboid: Number of Cuts (n) | Number of Regions (P) | 0 | 1 | 1 | 2 | 2 | 4 | 3 | 8 | 4 | 15 | 5 | 26 | n=1n=2n=3 n=4n=5

The sequence of the points seem to show a function that is too flat at close to the origin to be a parabolic function, perhaps it is a cubic function:
If the relationship is a cubic function, then the standard equation would equal:
P(n)=an3+bn2+cn+1
(The +1 is t make when n=0, P(n)=1, because, when there are no cuts, there is still one region.)

P(1)= a+b+c+1=2 a+b+c=1 P(2)=8a+4b+2c+1=4
8a+4b+2c=3

P(3)=27a+9b+3c+1=8
27a+9b+3c=7

8a+4b+2c=3
-2(a+b+c=1)
6a+2b=1 b=0.5-3a 27a+9b+3c=7
-3(a+ b+ c=1)
24a+6b=4
12a+3b=2

12a+3(0.5-3a)=2
3a=0.5
a=1/6

6(1/6)+2b=1 b=0 So: c=5/6 The Equation would be:
P(n)=(1/6)n3+(5/6)n+1
P(n)=(1/6)(n3-3n2+8n-6)+0.5(n2+3n+2)+(n+1)
Graphing this on the same plane as the points, and the equation proves true, as it contains every y-value for every n whole number equal to or greater than zero:

If the domain is extended, the equation still proves true.

So if P(5)=26, both from the reclusive and the conjectured equation, and

Show More

Essay On Black And White Gallery

Dallas Museum Experience

Hurst Vs Parthenon Frieze Analysis

Nt1310 Unit 1 Assignment

African Art Vs Native American Art

Art History Comparison Paper

Historical Criticism In John Steinbeck's Of Mice And Men

The Judgment Of Paris By Lodovico David

Yoko Ono Cut Piece Analysis

Essay about Sonny's Blues