Number Stairs.
Extracts from this document...
Introduction
Number Stairs
Introduction
In this project I will investigate the relationship between the stair total and the position of the stair shape on the grid for three step stairs. Then I will move on to investigating the relationship between the stair totals and other step stairs on other number grids. Below is a 10x10 grid with a three step stair highlighted in yellow and another in red.
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  16  16  17  18  19  20 
1  2  3  4  5  6  7  8  9  10 
Strategy
First we must define the variable. We will first choose the number in the bottom left hand corner of the step stair. I will call it n. I will then move the stair sequence systematically, with n increasing in tens, starting from five. This is because it is impossible to have a three step stair where n is any of the nonhighlighted numbers below.
Table of results
n  5  15  25  35  45  55  65  75 
total  74  134  194  254  314  374  434  494 
Middle
We can now put these numbers into the formula and find that
6n+44
is the formula to find the sum of any three step stair on a 10x10 grid.
To prove this I will put the algebraic values for each square in the three step stair.
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  n+20  34  35  36  37  38  39  40 
21  22  n+10  n+11  25  26  27  28  29  30 
11  12  n  n+1  n+2  16  17  18  19  20 
1  2  3  4  5  6  7  8  9  10 
We can now see that there are six squares in the stair (6n) and the numbers, which show how much they increase n by, add up to 44 (+44)
Step 2
Now we can begin to either vary the size of the grid or vary the size of the stair shape.
I will vary the size of the grid first because this is the easier option as there will still only be six numbers to add up in the three step stair.
Conclusion
Total = as^3 + bs^2 + cs
where s = The number of steps on the stair
In order to work out this formula a simeltaneous equation must be used.
Firstly I will take the equations used for the 2 and 3 step stair.
3n + 11 6n + 44
These equations can then be manipulated and added to the formula, and then be worked out.
11 = 8a + 4b + 2c * 3
44 = 27a + 9b + 3c *2
33 = 24a + 12b + 6c
_
88 = 54a + 18b + 6c
55 = 30a + 6b
55/30 = 1 5/6
From here we can then work out the value of c
11 = 8a + 4b + 2c * 27
44 = 27a + 9b +2c *8
297= 216a + 108b + 54c

352 = 216a + 72b + 24c
55 = 36b + 30c
55/30 = 1 5/6
To test to see if the formula works I will first convert the fractions to top heavy ones.
1 5/6 = 11/6 1 5/6 = 11/6
Now the fractions will be placed into the formula.
(11/6*3^3) + (11/6*3) = 44.
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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