# Investigate the relationship between different stairs placed in different places in a number square, which is numbered from one to one hundred.

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Introduction

## Introduction

In this project I am going to investigate the relationship between different stairs placed in different places in a number square, which is numbered from one to one hundred. I will be trying to find a formula to get a total of any stair any place.

Example

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 | 20 |

11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

This is a 3-step stair

Middle

23 23+13+14=50

1314

33 33+23+24=80

2324

43 43+33+34=110

3334

I have found out that when you move the stairs up by one the product of the two-step stair goes up by 30. This also means that when you move the two-step stair down by one the product will go down by -30. I have also found out the nth term for all two-step stairs that 3n+11 will equal the sum off all the numbers in the stair. I found out this by multiply the number in the bottom left hand corner by how many numbers that are in the stairs, then I added how much is needed to make the product correct, I shall use this to try and find out the other nth terms. I will now test my formula.

45 45+35+36=116

3536

To do the nth term I have to times 3 by 35 then plus 11, which equals 116

I will now move on to a three-step stair.

82

7273 82+73+72+64+63+62=416

626364

81

7172 81+72+71+63+62+61=410

616263

80

7071 80+71+70+62+61+60=404

606162

I have found out that every time you move the three-step stair one to the left the product of the three-step stair goes down by six.

Conclusion

53

4344

333435 53+43+44+33+34+35+23+24+25+26+13+14+15+16+17=415

23242526

1314151617

63

5354

434445 63+53+54+43+44+45+33+34+35+36+23+24+25+26+27=565

33343536

2324252627

73

6364

535455 73+63+64+53+54+55+43+44+45+46+33+34+35+36+37=715

43444546

3334353637

I have found out that every time you move the 5-step stair up by 1 the product goes up by 150, this means that when you move the 5-step stair down by one the product goes down by 150. I have also found out the nth term for this which is 15n+220, I shall now test this.

64

5455

444546 64+54+55+44+45+46+34+35+36+37+24+25+26+27+28=580

34353637

2425262728

To do the nth term I have to times 15 by 24 then plus 220 which equals 580. The nth term is correct.

Nth term | How many steps | Difference 1 (how much you plus) | Difference 2 (how much you times) |

1n+0 | 1 | ||

3n+11 | 2 | 22 | 2 |

6n+44 | 3 | 33 | 3 |

10n+110 | 4 | 44 | 4 |

15n+220 | 5 | 5 |

With this information I can find out any nth term. I shall now try and find out more nth terms.

Nth term | How many steps | Difference 1 | Difference 2 |

21n+165 | 6 | 55 | 6 |

28n+231 | 7 | 66 | 7 |

36n+308 | 8 | 77 | 8 |

45n+398 | 9 | 88 | 9 |

55n+495 | 10 | 99 | 10 |

Conclusion

Overall I have found out the nth terms for a 1 to a ten-step stair and some patterns with the nth term. I have also found out how much a 1 to a 5-step stair product changes if you move it up, down, left and right.

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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