# Number stairs

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Introduction

MATHEMATICS COURSEWORK- NUMBER STAIRS My investigation is based on number stairs. An example of a number stair is below: This is a 3 step stair because both the length and the width of the stair are 3 steps For the first part of my investigation this 3-step stairs (above is) is going to be placed on different number grids such as a 10 by 10 or 9 by 9 Number Grid etc Secondly for my first part of my investigation I also need to find a formula for each grid. These formulas must be able to work out the stair total for the 3-step on a number of different size grids such as 10 by 10 or 9 by 9 etc A stair total is all the values added together in the 3-step stairs After I have found a formula for a couple of number grids I am going to work out a general formula for any grid size possible. Below is a stair drawn on a 10 by 10 Number Grid: On the 10 by 10 Number Grid (on the previous page) a 3-step stair is highlighted The total of the numbers inside the stair shape is: 91+81+82+71+72+73=470 The stair total for this 3-step stair is 470 PART 1 For other 3-step stairs, investigation the relationship between the stair total and the position of the stair shape on the gird To start my investigation I am going to start by using a 10 by 10 Number grid below: I have highlighted a 3-step stair above on my 10 by 10 Number Grid The total of the numbers inside the stair shape is: 91+81+82+71+72+73=470 The stair total is all the six numbers in 3-step stair added together so for this 3-step stair the stair total is 470 On the right is a portion of the 10 x 10 grid squares and there is 6 boxes which are representing the numbers 91, 81, 82,71,72,73 and these are the 3-step stair From this diagram of ...read more.

Middle

+ 52 T= 174+52 T=226 The total for all the stair values added together without a formula is= 53 + 41 + 42+ 29+ 30 + 31= 226 My formula to find the total values in a 3-step stair on a 12 by 12 grid is correct. This is because all the stair values in a 3-step stairs added together on a 12 by 12 Number grid gives 226 and I also get 226 when I use my formula. This means I have proved for my formula to be correct If I closely observe the results from the 3-step stairs from the 3 number grids, such 10x10, 11x11 and 12x12 I can see there is a constant number that is consistent, which is [4] Below is a table that I have drawn by using the constant number [4] GRID SIZE ALGEBRA FORMULA 3 x 3 6 x + 16 4 x 4 6 x + 20 5 x 5 6 x + 24 6 x 6 6 x + 28 7 x 7 6 x + 32 8 x 8 6 x + 36 9 x 9 6 x + 40 10 x 10 6 x + 44 11 x 11 6 x + 48 12 x 12 6 x + 52 13 x 13 6 x + 56 14 x 14 6 x + 60 From the table above and using my results from the investigations Number Grids 10 x 10, 11 x 11 and 12 x 12 I can see a pattern emerging The value in the algebra formula increases by 4 every time the size of the grid box increases by 1. An example of this is the 10 by 10 Number Grid, where the 3-step formula is 6x-36 where x is the number in the bottom left hand square, then I increased the grid size by 1 so 11 x 11 and using the same 3-step stair approach the formula is 6x-40, etc This proves that my theory of the ...read more.

Conclusion

The investigation and tasks carried out in Part 1 and Part 2 show clearly illustrates the common relationship when using the general formula THE COMPLETE AND ENTIRE FORMULA The aim of The Complete And Entire Formula is to calculate the total value of the numbers in a step stair. From my investigations and the results of my table I can state the hypothesis as: Each time the step stair increases so does the amount of squares within the step, which is called the formula of triangular numbers. Also as the size of the grid increase by x number of squares the value of the squares total will also change again known as the triangular numbers. Below is a representation of triangular numbers: P = 1 3n P = 2 6n P = 3 10n P = 4 15n P = 5 21n From the triangular numbers diagram the 1st set of numbers are referred to as the "a" numbers (1,4,10,20,35,56) 2nd set as the "c" numbers (1st difference - 3,6,10,15,21,28) 3rd set as the "d" numbers (2nd difference - 3,4,5,6,7) Last set "e" numbers (common difference all the 1's). Then P is referred to as the step index. Consequently if P = 5 then there will be 21 squares (1st difference) in the step stair, making it a 6-step stair as shown above on the next page: ` I have found another pattern. If I look at the '1st difference' column, I noticed that all the numbers are triangular numbers. This could be help to find the second part of the formula '1/2 n� + 1/2 n' The "Complete & Entire Formula" (which I was not able to find) would have demonstrated that no matter what the size of the step stair (such as the number of squares), the position of the step stair (for example 1,11,12,21,22,23 or 81,82,83,91,92,93) and the grid size (e.g. 10x10, 23x23) the result will always be the total of the numbers in the step stair. This relationship is known as the "triangular numbers"+ ?? ?? ?? ?? Maryam Ahmad Mathematics GCSE Coursework-Number Stairs ...read more.

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