# For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. To start the investigation a 10x10-numbered grid square is used as illustrated below in table 1:

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Introduction

Mill Hill County High School Year 11 Mathematics GCSE Coursework EDEXCEL 2003, SYLLABUS 1387/1388 F, I & H Tejesh Patel Class 11H Assignment Part 1 Below is a 10x10 number grid: The total on the numbers coloured in blue = 90 (i.e. 1+11+12+21+22+23) Therefore the stair total in this 3-step stair = 90 Part 1 Objective: For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. To start the investigation a 10x10-numbered grid square is used as illustrated below in table 1: From the complete 10 x 10 numbered grid square, we use part of it to carry out our initial investigation, for example the grid box on the right shows a slice of the 10 x 10 gird square, i.e. 6 boxes representing the numbers 1,11,12,21,22,23 (The 3-step stair) From this basic numbered square that looks like stairs or steps we can start to establish if there is a pattern. If a pattern is found then we can use an algebra equation to represent this pattern and use the equation for a 10 x 10 numbered Grid Square. By using the 3-step stair example we know there are [6] squares and lets assume in a 3-step stair the bottom grid box is equal to [x], therefore in our 3-step stair x = 21 Using the values in algebra the formula(s) would look like this: The 1st square = 1 then the formula is x - 20 = 1 The 2nd square = 11 then the formula is x - 10 = 11 The 3rd square = 12 then the formula is x - 9 = 12 The 4th square = 21 it is simply just x = 21 The 5th square = 22 then the formula is x + 1 = 22 The 6th square = 23 then the formula is x + 2 = 23 The above algebra equations are shown below in our 3-step stair: x - 20 x ...read more.

Middle

Therefore the algebra equation starts to look like this: 10 x - 10(n) and the next step is to test the equation; we will use 15 as the grid size. n = 15 and our 4-step grid numbers are 1,16,17,31,32,33,46,47,48,49 as shown below and x = 46: 1 16 17 31 32 33 46 47 48 49 Using the equation 10 x - 10(n) we can use the above values to see if the equation works (10 x 46) - (4 x 15) = 400 To check the equation is correct we added all the numbers: 1+16+17+31+32+33+46+47+48+49 = 320 Therefore 400 <> 320 and our equation is incorrect. The value from the equation is lower that the correct result, therefore we need to reduce n, and we can start this by 1 i.e. (n - 1). For example 10 x - 10 (n - 1) we can again test this using the above 3-step grid. (10 x 46) - (10 x (15 - 1) 460 - 140 = 320 320 = 320 therefore the result from our equation is the same as it is by adding up the numbers. From the steps we have taken we have formulated an algebra equation which when tested, gives us positive results in every case: THE GENERAL FORMULA IS: Five step stairs: 10 x 10 Grid 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 x-40 x-30 x-29 x-20 x-19 x-18 x-10 x-9 x-8 x-7 X X+1 ...read more.

Conclusion

+ 1] X (x) - [1 + (p - 1)/6 (p2 + 4p + 6)] X (n - 1) Where p is the number in a triangular grid, x = the value of the bottom left grid in the step stair and n = the size of the grid Taking our 16x16 grid example we can calculate the total number of the value for the step stair: P = 5 x = 133 n = 16 [[1/2 (5 x 5) + ((3 x 5)/2) + 1] X (x)] = 21x - [1 + ((5 - 1) ((5 x 5) + (4 x 5) + 6))/6 x (n - 1) = 35 (n - 1) = 21x - 35 (n - 1) = (21 x 133) - (35 x 15) = 2793 - 525 = 2268 To test the answer we add the numbers: 53+69+70+85+86+87+101+102+103+104+117+118+119+120+121+133+134+135+136+137+138 = 2268 This demonstrates that the universal formula works for any step stair, any numbered grid size and combination of numbers: We can do one more test using the same step stair, but starting the step at 104 as shown on the side Therefore: p = 2 because there are 6 squares i.e. 6n and x = 136 and n = 16 Using the universal formula we can calculate: [[1/2 (2 x 2) + ((3 x 2)/2) + 1] X (x)] = 6x - [1 + ((2 - 1) ((2 x 2) + (4 x 2) + 6))/6 x (n-1)=4(n - 1) = 6x - 4 (n - 1) = (6 x 136) - (4 x 15) = 816 - 60 = 756 Add the numbers: 104+120+121+136+137+138 = 756 The formula shows that no matter what the size of the step stair (i.e. number of squares), the position of the step stair (e.g. 1,11,12,21,22,23 or 81,82,83,91,92,93) and the grid size (e.g. 10x10, 25x25) the result will always be the total of the numbers in the step stair. This relationship is known as the "triangular numbers" GCSE Mathematics Course work Tejesh Patel, Class 11h Page 21 GCSE Mathematics Course work Tejesh Patel, Class 11h Page 2 ...read more.

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