# Maths Number Stairs

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Introduction

Maths Investigation Introduction This investigation is called number stairs. I am going to investigate the relationship between the stair total and the position of the stair shape on the grid. Secondly I will do this by finding formula's for each step stair on a 10x10 grid. I will then look for a general formula to work out the position. I will then see if my general formula works on other grids, and then I will try to find one whole general formula to work out anywhere on any step stair or grid. I will now begin to find out the relationship I will start by looking for patterns on a 2 step stair and then moving on. 2 Step Stair Total = 14 Total = 58 Total = 47 Total = 113 Finding the formula I will now try to find the formula for my 2 step stair. I have noticed patterns. So I have decided to change my numbers into letters and making my smallest number n. Like this. Now I will take all my n's and add them together, to hopefully give me a formula. So n(n+1)+(n+10) = Hopefully my formula for a 2 step stair is 3n+11. I will check if my formula is correct by predicting the 5th term total. I will do this by substituting any value with n. 1 will use this stair. As an example Total = 152 one I have already used 3n+11 3 x 1 + 11 = total or 14 3 x 47 + 11 = total 14 = total or 14 152 = Total So I am correct my formula works. 3 step stair Total = 50 Total = 116 Total = 230 Total = 362 Finding the formula As before I will do the snme method to find my formula. So n+(n+1)+(n+2)+(n+10)+(n+11)+(n+20) = is my formula I will check to see if it works in same way as before. ...read more.

Middle

+ (n +2)+(n+3)(n+12)+ (n+13)+(n+14)+(n+24)+(n+25)+(n+36) = Total = 220 5 Step stair = n + (n+1) + (n +2)+(n+3)+(n+4)+(n+12)+ (n+13)+(n+14)+(n+15)+ (n+24)+ (n+25)+(n+26)+(n+36)+(n+37)+(n+48) = Total = 845 The formulas 2 Step Stair 3n + 13 3 Step Stair 6n +52 4 Step Stair 10n+130 5 Step Stair 15n + 260 I have noticed that the first part follows the same sequence as when on the 10x10 grid. 3, 6, 10, 15, follow the rule of the triangular numbers so there would be no need to find a formula for a them. So I can move straight to the other part. 0 12 52 130 260 13 39 78 130 26 39 52 13 13 Because of the three level of difference, again I will use the cubic formula N = step stair My list of formulas 1)a + b + c + d = 0 2)8a + 4b + 2c + d = 13 3)27a + 9b + 3c + d = 52 4)64a + 16b + 4c + d = 130 5)125a + 25b + 5c + d = 260 6)216a + 36b + 6c + d = 455 Take away 2) from 1) Take away 3) from 2) 2) 8a + 4b + 2c +d =13 3)27a + 9b + 3c + d = 52 1) a + b + c + d = 0 2) 8a + 4b + 2c +d =13 a) 7a + 3b + c = 13 b)19a + 5b + c = 39 Take away 4) from 3) Take away 6) from 5) 4)64a + 16b + 4c + d = 130 6)216a + 36b + 6c + d = 260 3)27a + 9b + 3c + d = 52 5)125a + 25b + 5c + d = 130 c)37a + 7b + c = 78 d)61a + 9b + c = 130 Take away b) from a) Take away d) ...read more.

Conclusion

You can also rearrange the formula to find which ever variable you want. Now I will prove that my formula works On a 3x3 grid 3 step stair N = 3 g = 3 x = 1 4 x 27 + 1 x 9 - 1 x 3 6 2 6 18 + 9 - 1 = 22 Total = 22 My formula seems to be working 2 step stair = 4 x 8 + 5 x 4 + 15 - 3 - 1 x 2 6 2 6 = 16 + 10 +11 3 3 Total = 19 = 19 I am correct both times my formula works Proving it on a 5x5 4 step stair using same method again T = 5+ 1 4 + 2 4 + 3(2) - 5 - 1 4 6 2 6 T = 6 x 64 +2 x 16 + 1 x 4 6 2 6 T = 64 + 18 + 0 Total = 80 T = 80 I am correct as you can see I have proved that my formula works three times. Proving it on a 7x7 grid 3 step stair T = 7+1 3 + 3 3 + 3(3) - 5 -1 3 6 2 6 T = 8 x 27 + 3 x 9 + 3 x 3 6 2 6 T = 36 + 27 + 3 2 2 Total = 51 Total = 51 As you can see again the formula works. Conclusion In this coursework there were two parts to it. I have successfully completed both parts. In part 1 I approached it in a different way to part 2. I found that my formula for a 10x10grid only works for 10x10. 1 also found patterns within the step stairs g = grid size = This increases by grid size Smallest Number This diagram above shows the patterns and variables which are grid size, smallest number, total, step number. These patterns helped together by using the cubic and quadratic function helped me and so together I found my general formula. By Mohamed Ibrahim 11h ...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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