# Number stairsMy aim is to investigate the relationship between the stair total and the position of the stair shape on the grid for 3 step

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Introduction

Jaie Wilde

Number stairs

My aim is to investigate the relationship between the stair total and the position of the stair shape on the grid for 3 step stairs and to investigate further the relationship between the stair totals and other step stairs on other number grids.

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 |

To begin I will try to identify a pattern. Taking my example I will take all the three stair steps on the bottom row and see if there are any similarities between the results.

- 1+2+3+11+12+21=50
- 2+3+4+12+13+22=56
- 3+4+5+13+14+23=62
- 4+5+6+14+15+24=68
- 5+6+7+15+16+25=74
- 6+7+8+16+17+26=80
- 7+8+9+17+18+27=86
- 8+9+10+18+19+28=92

Therefore I can now find a formula. The general term for an arithmetic sequence is Un=ab+c. The terms go up in sixes and this tells me that the nth term will include 6 lots of n or 6n. For the first term n=1, so 6n=6. But the first term is 50 which is 44 more than 6n. This suggests that the formula is 6n+44.

Trying a few values of n will help prove that my formula is correct.

(6 multiplied by 1) +44 =50.

(6 multiplied by 2) +44 =56.

(6 multiplied by 3) +44 =62.

(6 multiplied by 4) +44 =68.

(6 multiplied by 5) +44 =72.

I will now pick a random 3 step stair and test my formula.

76 | ||

66 | 67 | |

56 | 57 | 58 |

Step 1) Locate n which is in bottom left corner.

N=56

Step 2) Insert n into formula 6n+44.

Step3) (6 multiplied by 56) +44= 380 which equals the sum of the numbers.

The formula has worked.

With this formula I can now identify the sum of any 3 step stairs. I will now go on to try and identify similar formulas for bigger stair steps.

Middle

I will now test my formula by selecting a random 6 step stair.

72 | |||||

62 | 63 | ||||

52 | 53 | 54 | |||

42 | 43 | 44 | 45 | ||

32 | 33 | 34 | 35 | 36 | |

22 | 23 | 24 | 25 | 26 | 27 |

I have now successfully found formulas for 3, 4, 5 and 6 step stairs.

Number of stairs | formula |

3 | 6n+44 |

4 | 10n+110 |

5 | 15n+220 |

6 | 21n+385 |

From this table I have noticed that the numbers before n are triangular. Therefore I assume that the formula for a seven step stair will include 28n.

I will now test this on a seven step stair to see if this is true.

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 |

Also I have noticed that the number of stairs corresponds to the first term of the formula.

3 | 6n+44 |

4 | 10n+110 |

5 | 15n+220 |

6 | 21n+385 |

To get from 3 to 6 you multiply by 2.

To get from 4 to 10 you multiply by 2.5.

To get from 5 to 15 you multiply by 3.

To get from 6 to 21 you multiply by 3.5.

If I put this in a table:

1 | 2 | 3 | 4 | 5 | 6 |

1 | 1.5 | 2 | 2.5 | 3 | 3.5 |

The difference between the terms is ½ so my formula must start with ½ n. If I take term 4 I will half it to get two then I will add half to get to 2.5. This applies to all the terms. Therefore my formula is ½ n+ 0.5. This shows me what to multiply the number of step stairs by to find out the first term of the formula.

I will take an 8 step stair as an example to prove my formula.

If I put 8 in the formula it would be ½ of 8 which is 4. Then add 0.5 which is 4.5. This shows me that I have to multiply 8 by 4.

Conclusion

25 | ||||

19 | 20 | |||

13 | 14 | 15 | ||

7 | 8 | 9 | 10 | |

1 | 2 | 3 | 4 | 5 |

6 step stairs.

57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |

49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 |

41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |

33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |

17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

50 | |||||

42 | 43 | ||||

34 | 35 | 36 | |||

26 | 27 | 28 | 29 | ||

18 | 19 | 20 | 21 | 22 | |

10 | 11 | 12 | 13 | 14 | 15 |

I will start by taking a random 6 by 6 step stair. I can see that x = 10. If I write this out algebraically I would get: (x) (x + 1) (x+2) (x+3) (x+4) (x+5) (x+g) (x+g+1) (x+g+2) (x+g+3) (x+g+4) (x+2g) (x+2g+1) (x+2g+2) (x+2g+3) (x+3g) (x+3g+1) (x+3g+2) (x+4g) (x+4g+1) (x+5g)

I will now take a random 6 step stair from a 10 by 10 grid.

95 | |||||

85 | 86 | ||||

75 | 76 | 77 | |||

65 | 66 | 67 | 68 | ||

55 | 56 | 57 | 58 | 59 | |

45 | 46 | 47 | 48 | 49 | 50 |

If I wrote this out algebraically I find that it would be the same as in the 8 by 8 grid being: (x) (x + 1) (x+2) (x+3) (x+4) (x+5) (x+g) (x+g+1) (x+g+2) (x+g+3) (x+g+4) (x+2g) (x+2g+1) (x+2g+2) (x+2g+3) (x+3g) (x+3g+1) (x+3g+2) (x+4g) (x+4g+1) (x+5g).

If I simplify all this I would find that the equation would be 21x + 35g + 35. This represents the equation for a 6 step stair on any size grid.

I will now test the equation on a random 6 step stair from a 6 by 6 grid.

31 | |||||

25 | 26 | ||||

19 | 20 | 21 | |||

13 | 14 | 15 | 16 | ||

7 | 8 | 9 | 10 | 11 | |

1 | 2 | 3 | 4 | 5 | 6 |

x=1 and g=6. If I substitute these into the formula I will get (21 multiplied by 1) + (35 multiplied by 6) add 35. This will equal 266. If you add the numbers individually you will also get 266 which shows that the equation works.

Here is a table of my results

Number of steps in staircase | Equation |

3 | 6x+4g+4 |

4 | 10x+10g+10 |

5 | 15x+20g+20 |

6 | 21x+35g+35 |

I will now try and find an equation for any size stair on any size grid. The final product will be an equation like this: (part1)x + (part2)g + (part3).

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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