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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,

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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.


 Task 1

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.

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            1+2+3+11+12+12=50

...read more.

Middle


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28

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  28+29+38        95

65

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  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.

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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.

...read more.

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

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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

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this 8-Step Stair

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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

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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


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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

image00.png11-12-2002 07:25 PM

image01.pngimage02.png

image03.png

image03.png

(RE: Number Stairs (urgent) )

Oh, i forgot to mention:

x = number of steps of the stair
n = 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

...read more.

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