Number Grid Coursework.

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Number Grid Coursework

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Use the following rule: Find the product of the top left number and the bottom right number in the square. Do the same thing with the bottom left and the top right numbers in the square. Calculate the following difference between these numbers.

INVESTIGATE!

The first thing I'm going to do is work out the rule for a 10 x 10 grid. To do this I'm going to work out what the difference is between each row using 2 x 2, 3 x 3, 4 x 4, and 5 x 5 grids inside the main 10 x 10 one.

0 x 10 grid

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2 x 2 3 x 3

23 x 14 = 322 33 x 15 = 495

3 x 24 = 312 difference = 10 13 x 35 = 455 difference = 40

48 x 39 = 1872 82 x 64 = 5248

38 x 49 = 1862 difference = 10 62 x 84 = 5208 difference = 40

96 x 87 = 8352 88 x 70 = 6160

86 x 97 = 8342 difference = 10 68 x 90 = 6120 difference = 40

4 x 4 5 x 5

41 x 14 = 574 46 x 10 = 460

1 x 44 = 484 difference = 90 6 x 50 = 300 difference = 160

67 x 40 = 2680 62 x 26 = 1612

37 x 70 = 2590 difference = 90 22 x 66 = 1452 difference = 160

92 x 65 = 5980 95 x 59 = 5605

62 x 95 = 5890 difference = 90 55 x 99 = 5445 difference = 160

I can also use algebra to work out the differences.

2 x 2:

X

x + 1

x + 10

x + 11

(x + 10)(x + 1) => x² + 11x + 10

( x)(x + 11) => x² + 11x difference = 10

3 x 3

X

x + 1

x + 2

x + 10

x + 11

x + 12

x + 20

x +21

x + 22

(x + 20)(x + 2) => x² + 22x + 40

(x)(x + 22) => x² + 22x difference = 40

4 x 4

X

x + 1

x + 2

x + 3

x + 10

x + 11

x + 12

x + 13

x + 20

x + 21

x + 22

x + 23

x + 30

x + 31

x + 32

x + 33

(x + 30)(x + 3) => x² + 33x + 90

(x)(x + 22) => x² + 33x difference = 90

5 x 5

X

x + 1

x + 2

x + 3

x + 4

x + 10

x + 11

x + 12

x + 13

x + 14

x + 20

x + 21

x + 22

x + 23

x + 24

x + 30

x + 31

x + 32

x + 33

x + 34

x + 40

x + 41

x + 42

x + 43

x + 44

(x + 40)(x + 4) => x² + 44x + 160

(x)(x +44) => x² + 44x difference = 160

I will now attempt to find the nth term for this.

Grid size: 2 ? 2 3 x 3 4 x 4 5 x 5

10 40 90 160

\ / \ / \ /

st difference: 30 50 70

\ / \ /

2nd difference: 20 20

The first differences are not the same so we have to take a second difference.

The co-efficient of n is half the second difference. So the co-efficient of n is therefore 10 and as we had to take a second difference n will therefore be squared.

0n²

I now have to take 10n² away from the difference.

E.g. if n is 2 then 10n² is 40 so if I take 40 away from the difference of the 2 x 2 grid I get the 10n² part.
Join now!


2 x 2 : difference = 10 n = 2

0 x 2² = 40. 10 - 40 = -30

3 x 3: difference = 40 n = 3

0 x 3² = 90 40 - 90 = - 50

4 x 4: difference = 90 n = 4

0 x 4² = 160 90 - 160 = - 70

5 x 5: difference = 160 n = 5

0 x 5² = 250 160 - 250 = - 90

-30 -50 -70 -90

\ / \ / \ ...

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