Number Stairs: Totals and Positions

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Number Stairs: Totals and Positions

When I have worked out the formulas for 5 different size stairs on 3

different grid sizes, I will go on to investigate the relationship

between all of the formulas. From this I will be able to discover the

ultimate formula that will allow me to work out any stair size on any

size grid and in any position.

As any other formula I will need to predict and prove it, to check

that it is correct. But I also need to investigate if there is

anywhere on the grid that my formula will not work.

Lastly I will write a conclusion to cover in brief, what I have

discovered from this piece of coursework.

3 Step Stair on a 10 by 10 Grid

Firstly I will investigate if there is a relationship between a stair

total and its position on a 10 by 10 grid.

[IMAGE]This is a 3 step stair on a 10 by 10 grid. Its total is 50, I

worked this out by adding up all the numbers inside it.

1+2+3+11+12+21=50

This stair is in position 1. I got this number from the stair’s

bottom-left corner. Using this number it will be easier to understand

what stair I am talking about. In algebra terms n = 1. (n meaning

position number.)

The below stairs are all 3 step stairs on a 10 by 10 grid, but they

are in different positions. I have also added up their stair total.

[IMAGE]

The below grid shows the position number and its total, from this I

will be able to work out an nth term.

[IMAGE]The difference between each total is 6, so this means that the

nth term starts with 6n.

I did 6n and took the answer from the stair total (t). This left me

with 44. 44 goes on the end of the nth term, making the formula 6n+44.

[IMAGE]There is also another way to work out this formula without

working out the nth term. The diagram to the left shows a 3 step

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