# Step-stair Investigation.

Extracts from this document...

Introduction

William Murray

Step-stair Investigation

Cousework Submission 8th December 2003

For my GCSE Maths coursework I was asked to investigate the relationship between the stair total and the position of the stair shape on the grid. Secondly I was asked to investigate the relationship further between the stair totals and the other step stairs on other number grids. The number grid below has two examples of 3-step stairs. I will use Algebra as a way to find the relationship between the stair total and the position of the stair on the grid. I will use arithmetic and algebra to investigate the relationships between the grid and the stair further. The variables used will be:

Position of stair on grid = X

Sum of all the numbers within the stair = S

Step Size= n

Grid size= 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 first thing I will do is find the formula for all 3-step stairs on a size 10 grid.

I started off by making the bottom left hand number X. X is also the position of the stair on the grid. So in the diagram coloured red above X=15. I then added up the rest of the numbers in the three-step stair in terms of X. So 16= X+1, 17=X+2, 25=X+10 etc. The 3 step-stair in terms of X looks like this:

X+20 | ||

X+10 | X+11 | |

X | X+1 | X+2 |

If you simplify all the Xs and all the numbers you end up with this: 6X + 44, X+X+1+X+2+X+10+X+11+X+20=6X + 44. By investigating the formula above you will find that it is the formula for all 3-step stairs on a size 10 grid. I worked this out by adding together all the numbers in the 3-step stair and then using the formula to see if the formula comes up with the total of all the numbers in the 3-step stair.

Middle

11

12

13

14

1

2

3

4

5

6

7

By using the formula 10X+10g+10=S, I worked out the total of the numbers inside the blue area of the 4-step stair.

(10*18)+(10*7)+10= 180 + 70+ 10= 260

Then added up the numbers in the 4-step stair as I did before.

18+19+20+21+25+26+27+32+33+39= 260

The two examples above prove that the formula 10x+10g+10 calculates the total of the numbers inside the area covered by a 4-step stair on any grid size.

5 step stairs:

X+4g | ||||

X+3g | X+3g+1 | |||

X+2g | X+2g+1 | X+2g+2 | ||

X+g | X+g+1 | X+g+2 | X+g+3 | |

X | X+1 | X+2 | X+3 | X+4 |

By adding all the Xs, all the gs and all the numbers together I got:

X+X+1+X+2+X+3+X+4+X+g+X+g+1+X+g+2+X+g+3+X+2g+X+2g+1+X+2g+2+X+ 3g+X+3g+1+X+4g = 15X+20g+20. This is the formula for all 5-step stairs on any size grid.

To prove this formula works for all size grids and therefore works in general I drew two different sized grids and did the following calculations:

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 |

By using the formula 15X+20g+20=S, I worked out the total of the numbers inside the green area of the 5-step stair.

(15*25)+(20*10)+20=595

Then I added all the numbers in the 5-step stair to see if it came up with the same answer:

25+26+27+28+29+35+36+37+38+45+46+47+55+56+65=595.

21 | 22 | 23 | 24 | 25 |

16 | 17 | 18 | 19 | 20 |

11 | 12 | 13 | 14 | 15 |

6 | 7 | 8 | 9 | 10 |

1 | 2 | 3 | 4 | 5 |

By using the formula 15X+20g+20=S, I worked out the total of the numbers inside the turquoise area of the 5-step stair:

(15*1)+(20*5)+20=15+100+20=135

I then added up all the numbers in the 5-step stair to see if it gave the same number:

1+2+3+4+5+6+7+8+9+11+12+13+16+17+21=135

These two diagrams prove that 15X+20g+20=S on all sized grids.

6-step stairs:

X+5g | |||||

X+4g | X+4g+1 | ||||

X+3g | X+3g+1 | X+3g+2 | |||

X+2g | X+2g+1 | X+2g+2 | X+2g+3 | ||

X+g | X+g+1 | X+g+2 | X+g+3 | X+g+4 | |

X | X+1 | X+2 | X+3 | X+4 | X+5 |

By adding all the Xs all the gs and all the numbers up together I got this:

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+2g+3+X+3g+X+3g+1+X+3g+2+X+4g+X+4g+1+X+5g = 21X+35g+35.

To prove that this formula works I drew up two grids and used the formula to calculate the total of the numbers inside the 6-step stair and saw if it was the right answer on both grids.

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 |

I did this calculation to see if the formula works.

21X+35g+35 = (21*11) + (35*8) + 35 = 546

11+12+13+14+15+16+19+20+21+22+23+27+28+29+30+35+36+37+43+44+51= 546

31 | 32 | 33 | 34 | 35 | 36 |

25 | 26 | 27 | 28 | 29 | 30 |

19 | 20 | 21 | 22 | 23 | 24 |

13 | 14 | 15 | 16 | 17 | 18 |

7 | 8 | 9 | 10 | 11 | 12 |

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

Conclusion

(3-1) = 2

Σ Tr = T1 + T2 = 1+3 = 4

r = 1

So, for a 3 step stair the value of (blank)g + (blank) = 4. If you look up the formula for a 3-step stair is 6X+4g+4. I then proposed that the formula:

(n-1) (n-1)

n(n+1) +Σ Tr = g + Σ Tr =

2 r = 1 r = 1

Is the formula for any step stair on any sized grid.

I tried it on the 4 step stairs I had investigated before. The 3-step stair, the 4-step stair, the 5-step stair and the 6-step stair.

So, by replacing n with 3, for a 3-step stair I get this:

(3-1) (3-1)

3(3+1) + Σ Τr = T1 + T2 g + Σ Tr = T1+T2

2 r =1 r = 1

So by using this formula that I have explained above:

The formula gives: 6X+4g+4. T1 + T2 = 4, (1+3).

This is the formula for a 3-step stair on any size grid. The formula works.

4-step stair:

(4-1) (4-1)

4(4+1) + Σ Τr = T1 + T2 + T3 g + Σ Tr = T1+T2 + T3

2 r = 1 r = 1

So by using this formula that I have exlained above:

The formula gives: 10X+10g+10. T1 + T2 + T3 = 10 (1+3+6)

This is the formula for a 4-step stair on any size grid. The formula works.

5-step stair:

(5-1) (5-1)

5(5+1) + Σ Τr = T1 + T2 + T3 + T4 g + Σ Tr = T1+T2 + T3 + T4

2 r = 1 r = 1

By using the formula I have explained above;

The formula gives: 15X+20g+20. T1 + T2 + T3 + T4 = 20, (1+3+6+10)

This is the formula for a 5-step stair on any sized grid. The formula works.

6-step stair:

(6-1) (6-1)

6(6+1) + Σ Τr = T1 + T2 + T3 + T4 + T5 g + Σ Tr = T1+T2 + T3 + T4 + T5

2 r = 1 r = 1

By using the formula I have explained above:

The formula gives: 21X+35g+35. This is the formula for a 6-step stair on any sized grid. The formula works.

I have now proved that the formula I found works for any step stair size on any size grid.

(n-1) (n-1)

n(n+1) +Σ Tr = g + Σ Tr =

2 r = 1 r = 1

This concludes my first coursework submission. Submitted on:

8th December 2003.

William Murray

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