# Number Stairs

Extracts from this document...

Introduction

GCSE Coursework – Number Stairs Investigation

Part 1

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 |

This is a 10 x 10 size grid with a 3-stair shape in gray. This is called the stair total.

The stair total for this stair shape is 25 + 26 + 27 + 35 + 36 + 45 = 194.

To investigate the relationship between the stair total and the position of the stair shape, I will use the far-left bottom square as my stair number:

This is always the smallest number in the stair shape. It is 25 for this stair shape.

Now , I'm going to translate this 3-stair shape to different positions around the 10 x 10 grid:

44 | ||||

34 | 35 | |||

24 | 25 | 26 |

Middle

The stair-total for this stair shape is 4 + 5 + 6 + 14 + 15 + 23 = 68

This table summarizes these results :

Stair number | 24 | 25 | 26 | 25 | 3 | 4 |

Stair Total | 188 | 194 | 200 | 206 | 62 | 68 |

In order to find a formula which give the stair total when I am given the stair number,

I am going to put the stair number as the position and the stair total as the term for the

sequence:

Position | 24 | 25 | 26 | 27 | 3 | 4 |

Term | 188 | 194 | 200 | 206 | 62 | 68 |

Difference | +6 | +6 | +6 |

I have noticed that there is an increase of 6 between two consecutive terms in this

arithmetic sequence. Therefore the term rule must be 6 x Position± number .

Position (n) | 24 | 25 | 26 | 27 | 3 | 4 |

Term ( T ) | 188 | 194 | 200 | 204 | 62 | 68 |

6n | 144 | 150 | 156 | 162 | 18 | 24 |

+ or – | +44 | +44 | +44 | +44 | +44 | +44 |

We can notice that the term is always 6 times the position, Add 44 .

Conclusion

To check if this formula works for other stair numbers, I will try another numbe 55

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 |

Stair-total = 6n + 44

( using the formula ) = 6(55) + 44

= 330 + 44

= 374

Stair-total = 55 + 56 + 57 + 65 + 66 + 75

( by adding ) = 374 ✓

This shows that my formula must work for all stair-numbers in 3-stair shapes on the 10 x 10 grid.

However, I have observed that you could just sub any number as 'n' into the formula and

you could still get a stair total even if that stair shape cannot actually be drawn on the grid.

For instance, we could substitute 'n' by 10 into the formula like this:

Tn = 6n + 44

= 6(10) + 44

= 60 + 44

= 104

However it is impossible to draw a stair shape with the bottom left-hand number as

10, simply because it would not fit.

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

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month