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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

...read more.

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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