# Number Stairs.

Extracts from this document...

Introduction

## Katie Muston

## Number Stairs

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 |

On the 10 by 10 number grid above I have drawn a stair shape. This is known as a three-step stair.

The total of the numbers inside the stair is:

55+45+46+35+36+37=254

So the stair total for this shape is 254.

The stair number for this shape is stair 35 as the number in the bottom left hand corner is 35.

Now we can investigate the relationship between the position of the stair shape on the grid and the stair total. The diagram below is stair 1 as the number in the bottom left hand corner is 1.

21 | ||

11 | 12 | |

1 | 2 | 3 |

As we see, the stair total for this 3-step stair1 on this 10 by 10-number grid is 50 as 1 + 2 + 3 + 11 + 12 + 21 = 50.

Below is the stair shape one square to the right of stair1. This stair is called stair2 as the number in the bottom left hand corner is 1. Stair2 is a translation of stair1 one square to the right.

22 | ||

12 | 13 | |

2 | 3 | 4 |

The stair total for this stair shape is 2 + 3+ 4 + 12 + 13 + 22 = 56.

Now we are going to find the step total for stair3 (a translation of stair2 one step to the right).

23 | ||

13 | 14 | |

3 | 4 | 5 |

The stair total for this stair shape is 3 + 4 + 5 + 13 + 14 + 23 = 62.

Now we can find the stair total for stair4 (which is a translation of stair 3 one square to the right).

24 | ||

14 | 15 | |

4 | 5 | 6 |

Middle

Stair Number: 1 11 21 31

### Stair total: 50 110 170 230

Difference: 60 60 60

I can now predict that the stair total for stair 41 is going to be 230 + 60=290.

Here is the stair to prove my prediction is correct.

61 | ||

51 | 52 | |

41 | 42 | 43 |

The stair total for this shape is 41 + 42 + 43 + 51 + 52 + 61 = 290.

We can therefore say that every time you move the shape one square down the total decreases by 60 and when you move one square up the grid then total increases by 60. Also when you move one square to the right of the grid the total increases by 6 and when you move one square to the left the total decreases by 6.

The reason the total increases by 60 when you move the shape one square up on the grid is because since there are 6 squares in a 3-step stair and each separate number increases by 10. 6 multiplied by 10 equals 60.

The reason the total increases by 6 when you move the shape one square to the right on the grid is because there are 6 squares in a 3-step stair and each separate number increases by 1. 1 multiplied by 6 equals 6.

We can now introduce algebra in order to find a common formula to find the stair total for and 3-step stair on a 10 by 10-number grid.

x+20 | ||

x+10 | x+11 | |

x | x+1 | x+2 |

Conclusion

7a+3b+c , 9a+5b+c , 37a+7b+c , 61a+9b+c

12a+2b , 18a+2b , 24a+2b

6a , 6a

I am now going to match the numbers in the cubic sequence formula above to the numbers that correspond to their positions in the co-efficient in front of g sequence.

6a = 1

a = 1/6

12a+2b = 3

12(1/6) + 2b = 3

2 + 2b = 3

2b = 1

b = 0.5

7a+3b+c= 3

7(1/6)+3(1/2)+c = 3

c= 1/3

a+ b + c + d = 1

1/6 + ½ + 21/3 + d = 1

d = 0

Therefore the equation for this part of the sequence will be :

(1/6 n3 + 1/2n2 + 1/3n) = coefficient of g

Section 3 : The Sequence For The Constant

The sequence for the constant will go exactly the same as the sequence for the co-efficient of g as all these values are the same. Therefore the equation for this part of the sequence will be :

(1/6 n3 + 1/2n2 + 1/3n) = the constant

Conclusion

##### Now that we have finished the three parts of the sequence we can put them together in order to make one equation for any size step-stair on any size grid.

(1/2 n2 + 3/2n + 1)x + (1/6 n3 + 1/2n2 + 1/3n)g + (1/6 n3 + 1/2n2 + 1/3n)= stair total

x means the number of squares in the stair shape.

g stands for grid size.

n is the equation number. ( If you don’t know the equation number it is the same number as the number of squares in the 2nd column from the left of the stair shape).

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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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