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Introduction

As part of my coursework I have chosen number grid.

According to the task, I followed four steps in 10   10 grid

1. A box is drawn round four numbers.
2. Find the product of the top left number and the bottom right number.
3. Do the same with the top right and bottom left number.
4. Calculate the difference between these numbers.

The results are noted in the table below.

 BOX MULTIPLIED NUMBER PRODUCT DIFFERENCE 4          5 4       15 60 10 14       15 5      14 70 26       27 26     37 962 10 36       37 27     36 972 81       82 81     92 7452 10 91      92 91     82 7462

From the worked examples I find that the difference between the product found for the diagnol square in the 10      10  is 10.

From these results I predict that the difference between the next products in a diagnol square of 2    2 will be 10in 10    10 grid.

 BOX MULTIPLIED NUMBER PRODUCT DIFFERENCE 31     32 31     42 1302 10 41     42 32     41 1312 66     67 66     77 5082 10 76     77 67     76 5092 89     90 89     100 8900 10 99     100 90     99 8910

Middle

4     37

148

90

24    25    25    26

7     34

238

34    35    36    37

21    22    23    24

31    32    33    34

21     54

1134

90

41    42    43    44

24     51

1224

51    52    53    54

56    57    58    59

66    67    68    69

56     89

4984

90

76    77    78    79

59     86

5074

86    87    88     89

The results show that the difference between the products in 4    4 square is 90.It is five times to the difference in 3    3 square.I predict that it will be true for next squares in 4    4 square in 10    10 grid.

 BOX MULTIPLIED NUMBER PRODUCT DIFFERENCE 7      8       9     10 17    18    19    20 7     40 280 27    28    29    30 10     37 370 90 37    38    39    40 33    34    35    36 43    44    45    46 33     66 2178 53    54    55    56 36     63 2268 90 63    64    65    66 46    47    48    49 56    57    58    59 46     79 3634 90 66    67    68    69 49     76 3724 76    77    78    79

The results show that the prediction was right.

What would be the difference in 5    5 square in this grid ?

 BOX MULTIPLIED NUMBER PRODUCT DIFFERENCE 4      5      6      7      8 14     15   16    17    18 24    25    26    27    28 4    48 192 160 34    35    36    37    38 8     44 352 44    45    46    47    48 11   12    13     14    15 21    22    23    24    25 31    32    33    34    35 11     55 605 160 41    42    43    44    45 15     51 765 51    52    53    54    55 26    27    28    29    30 36    37    38    39    40 46    47    48    49    50 26     70 1820 56    57    58    59    60 30     66 1980 160 66    67    68    69    70

The difference between the products is 160.

Conclusion

THE SEQUENCE OF NUMBERS:  10, 40, 90, 160,

 Position Number 1 2 3 4 Term 10 40 90 160 Difference 30 50 70 2nd Difference 20 20

The 2nd difference between the succesive terms is (+)20.So the nth term will be

DINO

Nth term = 20n – 10

=10

4th term  = 20     4 – 10

= 80 –10

= 70

5th term = 20    5 – 10

= 100 – 10

= 90

10th term =  20    10 – 10

= 200 – 10

= 190

 BOX MULTIPLIED NO PRODUCT DIFFERENCE x              x + 1 x(x + 11) x + 11 10 x + 10       x + 11 (x + 1) (x + 10) x + 10x + x +10 3    3 SQUARE x            x + 1       x + 2 x(x + 22) x + 22x x + 10    x +11      x + 12 (x + 2) (x + 20) x +22x +2x + 40 40 x +20    x + 21     x + 22 4    4  SQUARE x           x + 1      x + 2   x + 3 x + 10   x + 11    x + 12  x + 13 x(x + 33) x + 33x x + 20   x + 21    x + 22  x + 23 (x + 3) (x + 30) x +30x + 3x + 90 90 x + 30   x +31     x + 32   x + 33 x         x + 1      x + 2     x + 3     x + 4 x +10  x + 11    x + 12    x + 13   x + 14 x(x + 44) x + 44x x +20   x +21   x + 22     x + 23  x + 24 (x + 4) (x +40) x + 40x +4x + 160 160 x + 30   x + 31   x + 32   x + 33   x + 34 x +40   x + 41   x + 42    x + 43   x + 44

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