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• Level: GCSE
• Subject: Maths
• Word count: 1115

# 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 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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# Related GCSE Number Stairs, Grids and Sequences essays

1. ## Number stairs

look like: - The 1st square = 109 then the formula is x + 24 = 109 -The 2nd square = 97 then the formula is x + 12 = 97 -The 3rd square = 98 then the formula is x + 13 = 98 -The 4th square = 85

2. ## For other 3-step stairs, investigate the relationship between the stair total and the position ...

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

1. ## Maths Grids Totals

The formula for the 2 x 4 rectangle is 10(2-1)(4-1) = 10 x 1 x 3 = 30. This is correct. The formula for the 3 x 5 rectangle is 10(3-1)(5-1) = 10 x 2 x 4 = 80. This is also correct.

2. ## number stairs

+44 but this formula only works for a 10 by 10 grid. I will investigate the relationship between the step-total, step-number and the grid size. In the 10 by 10 grid I noticed that the numbers 1, 11 & 21 etc increases by 10.

1. ## Maths coursework. For my extension piece I decided to investigate stairs that ascend along ...

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2. ## Number Stairs - Up to 9x9 Grid

is 9 and this number, when taken away from the number going upwards, gives the digit of the number going left to right. E.g. From 45 to 36 the difference is 9. From 45 to 35 the difference is 10.

1. ## Number Stairs

+ 44 = STAIR TOTAL. Now that I have worked out the formula for the 10x10 grid I am going to use the formula in random staircases with random stair numbers. And those are: Stair number = 17 Stair total = (6x17) + 44 = 146, or alternatively, 17+18+19+27+28+37= 146.

2. ## Mathematics - Number Stairs

+ (n+2) + (n+8) + (n+9) + (n+16) = 6n + 36 8 9 10 11 12 1 2 3 T = 6n + 36 T = 6n + 44 4 5 3 Step-Staircase / Grid Width 9 19 10 11 1 2 3 n 1 2 3 4 5 T 46

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