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• Level: GCSE
• Subject: Maths
• Word count: 3711

# Number Stairs

Extracts from this document...

Introduction

I have been given a number grid that counts in ascending order from one to a hundred, beginning at the bottom left hand corner to end at the top right corner with the number one hundred. With this grid I have been given the task of investigating the relationship/s between the grid size, the stair size, the stair total and the ‘n’ number.

 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 35 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10

The step shape given above is a 3-step stair, simply because it consists of 3 steps.

The stair total is labelled as being the sum of all of the numbers in the stair shape:

24 + 25 + 26 + 34 + 35 + 44 = 212

The Stair total for this 3-step shape is 212

The ‘n’ number is defined as being the smallest number of the stair shape, in the grid above it is specified as ‘ 24’.

I will systematically work my way through this problem to find appropriate algebraic solutions to simplify the workings of the stair totals. In doing so, I will use different size grids and use different size stairs. Again, investigating relationships and discovering formulae for each problem I encounter.

I am going to start with a 10 by 10 grid with a 3-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 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 35 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 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 35 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10
 n T 1 56 2 62 3 68 4 74 5 80 6 86

Middle

Using the same method as the previous, I will develop a formula.

Again, you can see that the ‘T’ column is increasing by 6 each time the ‘n’ number is increased by 1.

The ‘T’ column increases by 6 each time, so we multiply the ‘n’ number by the increase:

• 6n, and then add the remaining amount to end up with the ‘T’ number.
• 6n + 36 = T

To check this formula I will calculate another sum to which I know the solution to:

 n T 1 42 2 48 3 54 4 60 5 66 37 258

Using another method, I will ensure that my formula is correct and if possible simplify the expression given.

 n+3G n+3G+1 n+3G+2 n+2G n+2G+1 n+2G+2 n+G n+G+1 n+G+2 n n+1 n+2

n+n+1+n+2+ n+G+n+G+1+n+2G=T

Simplifying the above equation using means of collecting up will give me:

6n+4G+4=T

For this grid, it would be:        6n+4*8+4=T

I have developed formulas for each grid (8 by 8, 9 by 9 and 10 by 10), I have come to the simple conclusion that the general formula 6n+4G+4=T applies to all grids as long as the stair number remains the same at 3 (3-step stair).

Developing further on the generic subject matter, I will find solutions on moving the ‘n’ number either along the x

Conclusion

n+5G+2

n+5G+3

n+5G+4

n+5G+5

n+5G+6

n+5G+7

n+5G+8

n+5G+9

n+4G

n+4G+1

n+4G+2

n+4G+3

n+4G+4

n+4G+5

n+4G+6

n+4G+7

n+4G+8

n+4G+9

n+3G

n+3G+1

N+3G+2

n+3G+3

n+3G+4

n+3G+5

n+3G+6

n+3G+7

n+3G+8

n+3G+9

n+2G

n+2G+1

N+2G+2

N+2G+3

n+2G+4

n+2G+5

n+2G+6

n+2G+7

n+2G+8

n+2G+9

n+G

n+G+1

N+G+2

N+G+3

n+G+4

n+G+5

n+G+6

n+G+7

n+G+8

n+G+9

n

n+1

n+2

N+3

n+4

n+5

n+6

n+7

n+8

n+9

Above I will develop a 6-step stair shape formula that will work with any grid:

N+n+1+n+2+ n+3+n+4++n+5+ n+G+ n+G+1+ n+G+2+ n+G+3+n+G+4+n+2G+ n+2G+1+n+2G+2+n+2G+3+ n+3G+n+3G+1+n+3G+2+n+4G+n+4G+1+n+5G=T

Using means of collecting up I can simplify this formula to:  21n+35G+35 = T

I will check that this formula is correct using the 10 by 10 grid:

 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 35 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10
 n T 1 406 2 427 3 448 4 467 5 486 6 505

Here I am using the formula to see whether it works…

1. 21*1+35*10+35=406
1. 21*2+35*10+35=427
1. 21*3+35*10+35=448

From the sums I have calculated above you can see that the formula works and is accurate as it follows my table completely.

I assume and believe that the formula will work with all problems that use a 6-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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