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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 non-highlighted 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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