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# 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.

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