‘T-TOTAL’ Add all the numbers up including the t-number.
50 51 52
60
‘T-NUMBER’ The number at the bottom of the ‘T’
69 algebraically Classified an ‘n’
I am investigating the relationship between the t-total and the t-number.
Pattern: I have noticed a pattern in my results that each time the t-number increases by 1 the t-total increases by 5.
I know that; As 5 is the most common difference the must be a ‘5n’ in the formula.
22 23 24
32
41
Un = 5n-63
T-total = (22+23+24+32+41)=142
T-number = 41
Un = (41*5) – 63 = 142
My formula 5n-63 or 5*t-number-63 works
Proof of formula:
n - 18
n - 17
25 26 27
35 n - 9
n - 19
44
n
n
n-9
n-17
n-18
+ n-19
----------
5n-63
I will now take my investigation further by using different grid sizes to investigate relationships between the t-total, t-number and the grid size.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
As before I noticed that the pattern increased by 5 each time the increase by 1. As the common difference is 5;
There must be a ‘5n’ in the formula.
5*22 = 110 so 110-70=40
I have discovered a rule, which is 5n-70
Prediction: Using a 10*10 grid, I predict for the T-total will be 155 when the T-number is 45
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
(24+25+26+35+45) = 155
(5*45)-70 = 155
I can see that there is a difference of 7
I will now try to find a formula, which relates the T-total and T-number in ANY grid size, and I will also make the T-total and T-number relate with the grid size to find an overall general formula.
T-number = n
Grid size = g
(10*10 grid)
n-2g
1 2 3
n-2g-1
12 n-2g+1
n-g 22
n
n
n-g
n-2g
n-2g+1
+ n-2g-1
-------------
5n-7g
So ‘5n-7g’ is the general formula, which works on any grid of any size connecting the T-total, T-number, and the grid size.
Now I will transform the T-shape; I will rotate it 180°
Hence investigate the relationship between the T-total, T-number, the grid size, and the transformation.
1 2 3 4 5 6 7 T-Number
8 9 10 11 12 13 14
15 16 17 18 19 20 21 17
T-Total
22 23 24 25 26 27 28 24
29 30 31 32 33 34 35 30 31 32
36 37 38 39 40 41 42
43 44 45 46 47 48 49
Pattern: I have noticed a pattern in my results that each time the t-number increases by 1 the t-total increases by 5.
I know that; As 5 is the most common difference the must be a ‘5n’ in the formula.
27
34
40 41 42
Un = 5n+49
T-Total = (27+34+40+41+42)= 184
T-Number =27
My formula 5n+49 or 5*t-number+49 works
Proof of Formula:
n
2
n+7
9 n+14
15 16 17
n+15
n+13
n
n+7
n+13
n+14
+ n+15
-----------
5n+49
I will now try to find a formula, which relates the T-total and T-number in ANY grid size, and I will also make the T-total and T-number relate with the grid size to find an overall general formula.
T-number = n
Grid size = g
(10*10 grid)
n
2
n+g
12 n+2g
21 22 23 n+2g+1
n+2g-1
n
n+g
n+2g
n+2g-1
+ n+2g+1
--------------
5n+7g
So ‘5n+7g’ is the general formula, which works on any grid of any size connecting the T-total, T-number, and the grid size.
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