1-2 1-8 5-18 13-32 25-50 41-72
= -1 = -7 = -13 = -19 = -25 = -31
-6 -6 -6 -6 -6
.
. . = 2n² – 6n + c
Formula: 2n² – 6n +
When n = 1
2 × (1²) – (6 × 1)
= 2 – 6
= - 4 + ? = 1
= - 4 + 5 = 1
c = 5
.
. . Formula for black squares = 2n² – 6n + 5
Finding the n th term for white squares:
Formula: an + b
0 4 8 12 16 20
4 4 4 4 4
Formula: + b
= 4
.
. . Formula = 4n + b
Formula: 4n +
When n = 1
1 × 4
= 4
= 4 + ? = 0
= 4 + - 4 = 0
b = - 4
.
. . Formula for white squares = 4n – 4
Finding the n th term for the total number of squares:
1 5 13 25 41 61
4 8 12 16 20
4 4 4 4
Formula: + bn + c
a = 2nd difference
2
a = 4/2
= 2
.
. . = 2n² + bn + c
Formula: 2n² + + c
1 5 13 25 41 61
2×(1²) 2×(2²) 2×(3²) 2×(4²) 2×(5²) 2×(6²)
= 2 = 8 = 18 = 32 = 50 = 72
1-2 5-8 13-18 25-32 41-50 61-72
= -1 = -3 = -5 = -7 = -9 = -11
-2 -2 -2 -2 -2
.
. . = 2n² – 2n + c
Formula: 2n² – 2n +
When n = 1
2 × (1²) – (2 × 1)
= 2 – 2
= 0 + ? = 1
= 0 + 1 = 1
c = 1
.
. . Formula for the total number of squares = 2n² – 2n + 1
Check:
formula for black squares + formula for white squares = formula for total no. of squares
2n² – 6n + 5 + 4n – 4 = 2n² – 2n + 1
= 2n² – 2n + 1 = 2n² – 2n + 1
Test:
When n= 7
Number of Black Squares = 2n² – 6n + 5
= 2(7²) – 9(7) + 5
= 98 – 42 +5
= 56 + 5
= 61
Number of Black Squares = 4n – 4
= 4(7) – 4
= 28 – 4
= 24
Total Number of Squares = 2n² – 2n + 1
= 2(7²) – 2(7) + 1
= 98 – 14 + 1
= 85
Geometric Proof:
After examining the shapes it was realised that it is possible to work out the total number of squares geometrically. If the total number of white squares on one side are counted and the total number of black squares on one side are counted in each cross-shape there will always be one more white square than black square.
Example: shape 5
There are 5 white squares on one side and 4 black squares on 1 side.
A square is drawn around the black squares and another square is drawn around the white squares. The areas of each of them are worked out. When they are added the total number of squares of the cross-shape will be calculated.
4² + 5² = 41
There are 41 squares in total of this cross-shape pattern.
The formula = n² + (n – 1)²
When multiplied out it becomes:
n² + (n – 1)²
= n² + (n – 1) (n – 1)
= n² + n² – 1n – 1n + 1
= 2n² – 2n + 1
This proves the formula.
Part 2:
Aim:
I shall investigate the link between the total number of cubes in a three-dimensional shape and the position of this shape in a geometric sequence. The shape used is a three-dimensional equivalent of the cross, used in the first part of my coursework.
Prediction:
As the three-dimensional shape increases in length, height and width, I predict an increasing number of cubes will have to be added. I predict that the total number of cubes will increase in proportion to n3, where n is the position of the shape in the geometric sequence. This is because as a shape increases in volume, its size increases in proportion to length3.
Finding the n th term for the total number of cubes:
Formula: an³ + bn² + cn + d
1 7 25 63 129 231
6 18 38 66 102
12 28 36 44
8 8 8
Formula: + bn² + cn + d
a = 3rd difference
6
a = 8/6
= 4/3
.
. . = 4/3n³ + bn² + cn + d
Formula: 4/3n³ + bn² + cn + d
1 7 25 63 129 231
4/3×(1³) 4/3×(2³) 4/3×(3³) 4/3×(4³) 4/3×(5³) 4/3×(6³)
= 4/3 = 32/3 = 36 = 256/3 = 500/3 =288
1-4/3 7-32/3 25-36 63-256/3 129-500/3 231-288
= -1/3 = -11/3 = -11 = -67/3 = -113/3 = -57
-1/3 -11/3 -11 -67/3 -113/3 -57
10/3 22/3 34/3 46/3 58/3
4 4 4 4
Formula: 4/3n³ + + cn + d
a = 2nd difference
2
a = 4/2
= 2
.
. . = 4/3n³ + 2n² + cn + d
Formula: 4/3n³ + 2n² + cn + d
-1/3 -11/3 -11 -67/3 -113/3 -57
2×(1²) 2×(2²) 2×(3²) 2×(4²) 2×(5²) 2×(6²)
= 2 = 8 = 18 = 32 = 50 = 72
-1/3+2 -11/3+8 -11+18 -67/3+32 -113/3+50 -57+72
= 5/3 = 13/3 = 7 = 29/3 = 37/3 = 15
8/3 8/3 8/3 8/3 8/3
.
. . = 4/3n³ – 2n² + 8/3n + d
Formula: 4/3n³ + 2n² + 8/3n +
When n = 1
[4/3 × (1³)] + [2 × (1²)] + [8/3 × 1]
= 4/3 + 2 + 8/3
= 6
= 6 + ? = 7
= 6 + 1 = 7
d = 1
.
. . Formula for the total number of cubes = 4/3n³ + 2n² + 8/3n + 1
Check:
When n = 3
4/3n³ + 2n² + 8/3n + 1
4/3(3³) + 2(3²) + 8/3(3) + 1 = 63
4/3(27) + 2(9) + 8 + 1 = 63
36 + 18 + 8 + 1 = 63
63 = 63
.
. . Formula is correct
