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• Level: GCSE
• Subject: Maths
• Word count: 4332

# Number Stairs

Extracts from this document...

Introduction

EDEXCEL 2003        CANDIDATE        SHEET        SYLLABUS
DISPATCH 1
1387/1388

MATHEMATICS        NUMBER STAIRS        F, I & H
GCSE

Look at the stair shape draw-n on the 10 by 10 Number Grid below,

 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 3-step stair.

The total of the numbers inside the stair shape is

25 + 26 + 27 + 35 + 36 + 45 = 194

The stair total for this 3-step stair is 194.

## Part 1

For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid.

## Part 2

Investigate further the relationship between the stair totals and other step stairs on other number grids.

I want to investigate the relationship between position of a 3-Step Stair and the total of the numbers inside the 3-Step Stair. In this investigation I
hope to find a formula which could be used to find out the total of the numbers inside a 3-Step Stair in any position on the 10 by 10 grid……. I will firstly carry out this experiment on a several 3-Step Stairs, starting with the stair with the smallest number at the bottom left corner  (1) and go on to 2,3,4…… until I can spot a pattern. By starting with  smallest number  will be easier to work with and by going systematically will enable me to spot any patterns that links them together so I will be able to generate a sequence. To generate a sequence, I will need a starting value and a rule to find the next term. My starting value will be the Stair Total of the first 3-step stair, and the rule to find the next term will be found by working through the Stair Total difference of first 3step stair and the second 3step stair, by subtracting the two totals.

 12 11 12 1 2 3

1+2+3+11+12+12=50

Middle

 38 28 29

28+29+38        95

 65 55 56

55+56+57      168

(Using formula 3(x)+11=n)

(x)    Workings     (t)                                                  Workings      (t)   Correct

26   26+27+36        89                 3(26)+11=      89    Correct

77   77+78+87      242                                                 3(77)+11=     242    Correct

28   28+29+38        95                                                 3(28)+11=       95    Correct

55   55+56+57      168                                                 3(55)+11=     168    Correct

All the results using the formula are correct, so I can conclude that this formula3(x)+11=(t) gives you the stair total of any 2-Step Stair in the 10 by 10 Grid which travels from bottom right to top left.

At this point I also notice that the 1st difference of the stair totals is equal to the number of cells inside the Stair. For example the difference of Stair Totals of 3-Step Stair is 6. and the number of cells inside the 3-Step Stair  is 6. I will draw a 3-Step Stair to show my finding more clear.

 1 2 3 4 5 6

As I can see from this 3-Step Stair it has 6 Cells or Blocks inside its grid. And the stair totals of 3-Step Stair have 1st  common difference which is 6 This Because the fact that the shape of the stair is actually triangular.

As you can see from this diagram the stair is triangular blocks with it is

cells just Shifted to the left

3-Step Stair

2tep

This means that if I find the formula for the number of cells inside the triangle shape, the first part of a general formula for all stair sizes will be found. And referring to my previous math lessons I know  the formula for the triangle number is. The height of the triangular shape (h), plus 1, divided by 2, multiplied by (h).     h(h+1)/2 . but instead of height (h). I will be using (s) as, Number of Steps of Stair. For example if the number of steps (s)=3, then 3((3+1)/2)gives you 6, which is exactly the number of cell inside 3-Step Stair.

Conclusion

The answers are correct and at this point I think I have found a universal formula for all the stair totals inthe 10 by 10 grid. Which is x(s(s+1)/2)+11s3-11s/6= t.

But to make sure it works for the other step stair totals I will test my formula on number of different sized step stairs in the 10 by 10 grid.

7-Step Stair

 61 51 52 41 42 43 31 32 33 34 21 22 23 24 25 11 12 13 14 15 16 1 2 3 4 5 6 7

1+2+3+4+5+6+7+11+12+13+14+15+16+21+22+23+24+25+31+32+33+34+41+42+43+51+52+61=644

The Stair Total of this 7-Step Stair is     =644

Using the formula= x(s(s+1))  +(11s3 - 11s)= t.

2                     6

(1)(7(7+1)) + (11(73))-(s11)=t

2                    6

(1)     (28)+    (616)       = 644

 2 3 4 5 6 7 8

this 8-Step Stair

 71 61 62 51 52 53 41 42 43 44 31 32 33 34 35 21 22 23 24 25 26 11 12 13 14 15 16 17 1 2 3 4 5 6 7 8

1+2+3+4+5+6+7+8+11+12+13+14+15+16+17+21+22+23+24+25+26+31+32+33+34+35+41+42+43+44+51+52+

53+61+62+71=960

The Stair Total of this 8-Step Stair is 960

Using the formula= x(s(s+1))  +(11(s3)– 11s)= t.

2                 6

1(8(8+1)  + (11(83)-11s)    =t

2                    6

1     (36)      +         924        =960

 81 71 72 61 62 63 51 52 53 54 41 42 43 44 45 31 32 33 34 35 36 21 22 23 24 25 26 27 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9

1+2+3+4+5+6+7+8+9+11+12+13+14+15+16+17+18+21+22+23+24+25+26+27+31+32+33+34+35+36+41+42+43+44+45+51+52+53+54+61+62+63+71+72+81=1365

The Stair Total of this 9-Step Stair is=1365

Using the formula=

x(s(s+1))  +(11(s3)– 11s)=t.

2                      6

1(9(9+1)) +   (11(s3)-11s) =t

2                       6

45              1320      =1365

 91 81 82 71 72 73 61 62 63 64 51 52 53 54 55 41 42 43 44 45 46 31 32 33 34 35 36 37 21 22 23 24 25 26 27 28 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+21+22+23+24+25+26+27+28+31+32+33+34+35+36+37+41+42+43+44+45+46+51+52+53+54+55+61+62+63+64+71+72+73+81+82+91=1870

Using the formula=

x(s(s+1))  +(11(s3)– 11s)=t.

2                      6

1(10(10+1)) +   (11(s3)-11s) =t

2                       6

55    +     1815  =1870

As I have proved from my test the formula works for every Step Stair in the 10 by 10 grid, and I can confidently conclude that x(s(s+1))  +(11(s3)– 11s)=t. works for

2                      6

any step stair that travels from bottom right to top left in the 10 by 10 grid.

You find the answer in time Dire? you said it was in for the 7th.
I found it the other day, sorry i didn't find your post sooner.
In the end i found something like:

(x(x+1)/2)n + (g+1)/6 (x-1) (x+1) ( x )

Report this post to a moderator

11-12-2002 07:25 PM

 (RE: Number Stairs (urgent) )Oh, i forgot to mention:x = number of steps of the stairn = nottom left number of the stair (place on grid)g = grid size. eg. on a 10 X 10 grid, g=10

The formula for 44 = h(height of the grid)  (11h3 – 11h)/6

The whole formula=  (x(x+1)/2)n + (11x3 – 11x)/6

And now I am going I am going to investigate the 4-Step Stair

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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