+6 +6 +6 +4? +6
For stair no.5 the total did not follow the pattern.I will check it’s total for a second time…
My first calculations were in-correct, the stair total is in fact 74
As you can see a strong pattern occurs.The total increases by six every time.From this I can conclude the first part of the equation must be 6n.To find the rest of the formulae I took a closer look at stair no. 1…
If 1 x 6 = 6
Then I must subtract this no from the total stair no. for 1 which is 50
50 – 6 = 44
Therefore the formulae must be…
6n + 44
Testing 6n + 44
I will now test the formulae 6n + 44…
Although this is the formulae for this row, there could be a sepearte rule for another horizontal row, further up the grid.This should be sufficient evidence to conclude whether the formulae works for all horizontal rows throughout the grid.
I will take a larger number for example in the 70’s to ensure I have a wide range of results.
+6 +6 +6 +6
We can see the formulae 6n + 44 also works for this, therefore the equation for a 10 x 10 grid horizontally will be ;
6 n + 44
Next I will see if the same rule works for number stairs going vertically.
Testing for the vertical rule
I took the stair no.s 1 ,11, 21 , 31, 41 …
Next I will need to work out the stair totals for each of the stair no.s
+60 + 60 + 60 +60
As you can see the pattern increases by 60 every time
Therefore we can conclude that the rule
6n + 44 definitely does not work
Although it seems to work for stair no. 1
The horizontal rule does cannot be applied to the vertical rule
60n -10
Instead I will test the formulae 60n – 10
I was lead to this formulae as the pattern increases by 60; therefore 60n seems correct
The second half of the equation is – 10 as
110 – 60 = 50
The first stair total works using this formulae!
I will now try 11
11 x 60 =660
660 – 10 = 650
This formulae definitely does not work
After much consideration , I have come to the conclusion that working out the formulae is beyond my mathematical knowledge.Although I can see a pattern
Re- ocurring.The totals increase by 60 every time.
To expand the investigation further I will now be using a 7 x 7 grid …
7 x 7 Grid
The 7 x 7 grid will look something like this…..
Horizontal
I will take the stair no.s 3, 4 , 5, 6, and 7
The totals are as follows…
6 6 6 6
As the difference is 6 the formula will be 6n
I will now find the second part of the rule
If 3 x 6 = 18
Then we need to subtract 18 from 42
= 24
The formulae should be 6n = 24
Testing
I will check if the formulae works with the other stair no.s
4 x 6 =24
24+24= 48
The formulae works for stair no.4
5 x 6 = 30
30+24= 54
The formulae works for stair no. 5
6 x 6 =36
36 + 24 = 60
The formulae works!
7 x 6 = 42
42 + 24 = 66
The formulae works for the whole row!
Therefore the formulae for the sequence
3 , 4, 5, 6, 7….
Is;
6n + 24
I will now try with another horizontal row to see if the results will be the same further up the grid.This will ensure that the formulae I am testing is correct for all horizontal rows across a 7 x 7 grid.
Taking the sequence 24, 25 , 26, 27 , 28…
The totals are seen below
6 6 6 6
The difference between the totals are again six.This is an advantage, are formulae is more likely to be correct.I will check the formulae against the sequence just in case.
Taking the formulae T= 6n +24
If …
24 x 6 +24=168
25 x 6 +24=174
26 x 6 +24=180
27 x 6 +24=186
28 x 6 +24=192
From this I can conclude the rule for a horizontal row in a 7 x 7 grid is ...
T= 6n + 24
Now I will try to find the vertical rule
Vertical
I will take the sequence 5 , 12 , 19 , 26 , and 33
42 42 42 42
The pattern shows an increase of 42
The equation should use 42 to multiply the subject
T=42n
If 5 x 42 = 210
To get to 54 we need to subtract 54 from 210
210 – 54=156
The formulae could be T= 42 n – 156
I will test the formulae using stair no. 12
It should total 96
12 x 42 = 504
504 – 156= 348
The formulae does not work
I will try another formulae using another sequence…
Sequence 6 , 13 , 20, 27 and 34
I will take stair no.s 6,13,20,27 and 34.
I then add up the totals for each stair no. and record it in the table below.
These were the results I found…
42 24 60 42
18 36 18
18 18
The difference is 18
I will check to see if the horizontal formulae applies to the vertical as well.
T=6n+24
Stair no. 6
6 x 6 +24= 60
The formulae works for stair no.6
I will now check, using the other stairs in the sequence.
13x 6 +24=102
The formulae works for stair no. 13
20 x 6 + 24 =144
The formulae does not work
Therefore it is not the formulae
Although I can identify the patterns I cannot see a formulae that works.
