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. A new formula must be created for the stairs with four levels or more and for those with 2 or less.
Two levels
n+(n+1)+(n+11)
=3n+11
25 + 26 + 35
= 86
(3x25) + 11 = 86
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=3x+11
One level
n = n
25 = 25
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 = x
Four levels
n+(n+1)+(n+2)+(n+3)+(n+10)+(n+11)+(n+12)+(n+20)+(n+21)+(n+30)
= 10+110
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=10x+110
No matter what x is replaced by the formula (10x+110) is always applicable when the stairs has 4 levels.
24 + 25 + 26 + 27 + 34 + 35 + 36 + 44 + 45 + 54
= 350
(10 x 24) + 110 = 350
Even though the formula has changed there are still some things that must stay the same. For example the formula must always have a multiple of x so that x can be replaced by the base number.
10x + 110 6x + 44
As the number of levels on the stairs increases then so will the numerical aspects of the formula before and after x.
5 levels = 15x + 220 7 levels = 28x + 616
6 levels = 21x + 385 8 levels = 36x + 866
The increase in x
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
main sequence – 1,3,6,10,15,21,28,36
1st difference – 2,3,4,5,6,7,8
2nd difference – 1,1,1,1,1,1
This shows that it is a triangular number sequence in the increase in x.
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