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# An investigation into the relationship between stairs size and the value.

Extracts from this document...

Introduction

An investigation into the relationship between stairs size

Middle

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1st stair: 25 + 26 + 27 + 35 + 36 + 45 = 194

If we represent this 1st stair in the form of n then an algebraic formula can be created.

n+(n+1)+(n+2)+(n+10)+(n+11)+(n+20)

= 6n+44

Therefore in terms of x and y, with x being the base number and y being the total of the stair the formula would be:

y=6x+44

No matter what x is replaced by the formula (6x+44) is always applicable.

25 + 26 + 27 + 35 + 36 + 45

= 194

(6x25) + 44 = 194

26 + 27 + 28 + 36 + 37 + 46

= 200

(6x26) + 44 = 200

These stairs are only one along from each other on the same line.  This formula applies to any 3 levelled stairs anywhere on the grid no matter where it is.

45 + 46 + 47 + 55 + 56 + 65

= 314

(6x45) + 44 = 314

78 + 79 + 80 + 88 + 89 + 98

= 512

(6x78) + 44 = 512

This formula however must be changed for a stair with a higher number of levels.  If the number of levels exceeds 3 then the formula (6x + 44) is incorrect.

Conclusion

The sequence of the triangular numbers comes from the natural numbers (and zero), if you always add the next number:

1
1+2=
3
(1+2)+3=
6
(1+2+3)+4=
10
(1+2+3+4)+5=
15
...

This diagram is identical to the number squares in the grid with the exception that they are not numbered and the diagram shows that the number of x in the formula is directly related to the number of boxes in the stair.  For example (6x+44) is the 3 level stair formula and in the diagram above there are six boxes in the 3 level stair.

The increase in x that is shown in the table shows that as the number of levels increase so does the amount of x.  The increase is by the same number.

e.g. when there are 3 levels the increase in x is by 3.  this shows that there is a sequence in the formula

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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