# Number Stairs

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Introduction

Number Stairs Introduction In this investigation I plan to find out how grid size affects the relationship between the total of a step stair and the position of the step stair shape on various sized number grids, e.g. 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 The shaded squares show a 3 step stair on a 10 x 10 grid, but I shall be investigating various sized stairs relationships between their positions and their totals on various grid sizes. The grid sizes I will be using in my investigation are as follows: 1. 10 x 10 2. 9 x 9 3. 8 x 8 The total for the stair is: 1 + 2 + 3 + 11 + 12 + 21= 50 The total for the stair is always worked out by adding up all the values inside of the stair shape. The position of the stair is 1. The stairs position is always worked out by the number in the bottom left hand corner. ...read more.

Middle

The nth term can be worked out by changing the numbers in the 6 step stair in to algebra by making the position number n, then relating all other numbers to the position number, as you up a level on the step stair you add the grid sizes number (in this case 10) to the number below it so is represented by g (grid size) e.g. n+5g n+4g n+4g+1 n+3g n+3g+1 n+3g+2 n+2g n+2g+1 n+2g+2 n+2g+3 n+g n+g+1 n+g+2 n+g+3 n+g+4 n n+1 n+2 n+3 n+4 n+5 The sum of this stair= (n) + (n+1) + (n+2) + (n+3) + (n+4) (n+5) + (n+g) + (n+g+1) + (n+g+2) + (n+g+3) + (n+g+4) + (n+2g) + (n+2g+1) + (n+2g+2) + (n+2g+3) + (n+3g) + (n+3g+1) + (n+3g+2) + (n+4g) + (n+4g) + (n+5g)= 21n + 35g + 35 (21 x 1) + (35 x 10) + 35= 406, this represents the 1st term. Using this formula I can now work out any term for a 6 step stair on a 10 x 10 grid, e.g. the 55th term: (21 x 55) + (35 x 10) + 35= 1540 Investigation for 7 step stairs on a 10 x 10 grid 61 51 52 41 42 43 31 32 33 34 21 22 23 24 25 11 12 13 14 15 16 1 2 3 4 5 6 7 62 52 53 42 43 44 32 33 34 35 22 23 24 25 26 12 13 14 15 16 17 2 3 4 5 6 7 8 63 53 54 43 44 45 33 34 35 36 23 24 25 26 27 13 14 15 16 17 18 ...read more.

Conclusion

'r' takes all values between these extreme values. 's' was previously used in the triangle number formula but I have now changed it to 'r' as this is the commonly used letter when using summation of a series. I have also changed the lettering because 's' represents step stair size and if 's' was used above and below the sigma symbol the highest value would always equal 0 making the formula incorrect. is used as it is the triangle number formula and the triangle number are what need to be added to make the required tetrahedral number. is used as the lowest value of r as it is the first triangle number so must be used as the start of every series. is used as the highest value because we always need the tetrahedral number that is one less than the step stair size e.g. Step Stair Size 'g' Number Tetrahedral Ranking 3 4 2nd 4 10 3rd 5 20 4th 6 35 5th I shall now test the formula using a 3 step stair: The added numbers at the end of the formula are always the same as the 'g' numbers making them tetrahedral numbers as well so they must be worked out in the same way: Added numbers= For example using a three step stair: Therefore the formula for working out any sized step stair's total on any sized grid is: I can now use this general formula to find out any sized stair on any sized grid, for example a 55 step stair on a 100 x 100 grid: (55) (56) = 1540 2 1540 x n= 1540n 27720 x g= 27720g This makes the formula 1540n+ 27720g+ 27720. 1 ...read more.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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