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maths stairs

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Introduction

Luke Griffiths

Number Stairs part 2

I am going to be investigating the relationship between stair totals using 3 by 3 size step stairs on different size grids. I am looking for an equation that will link the stair total to the size of the grid. I am going to do 3 different grid sizes and then predict the 4th. If I am successful I will use the formula. If I am unsuccessful I shall try again. I will use 1 as the corner squares, testing on 2, n as the term for the corner square, and g as the grid size. How I got the formula is explained in part 1. (Diagrams are above)

I

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Middle

I have worked out that if the corner square is 1 in a 4 by 4 grid the total will be.

1+2+3+5+6+9=26

I worked this out by adding all of the numbers inside the stair and finding the total.

I came up with the formula of 20+6n. As there is still the same number of ‘n’ in the diagram, the only change is that of the numbers. To test this I will do both on the

I predict that if the corner square is 2 in a 4 by 4 grid the total will be.

(6*2)+20=32

To prove I used the other method as well.

2+3+4+6+7+10=32

Therefore the 4 by 4 grid formula for a 3 step stair is 6n+20.

image02.png

I have now moved on to a 5 by 5 grid.

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Conclusion

span class="c3">However to prove this I predict that for the next grid increase, 6 by 6, the formula will be 6n+28.

Which would therefore mean that the formula would be (6*1)+28=34

image03.png

1+2+3+7+8+13=34

Therefore this constant increase is apparent and therefore I can use this to draw up a table of results, which is shown below.

image04.png

Now I need a general formula for a 3 step stair on any grid size. I will use g to represent the grid size.

The diagram is following. I have seen that the grid size is also the amount the n increases by from line to line. Therefore n+g will be the correct number proved by:

n=1, g=3

n+g=4

image05.png

4 is the number directly above n so therefore is n+g therefore the formula is correct.

image06.png

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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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