Number Stairs investigation.
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Introduction
Number Stairs investigation
The task of this coursework is to investigate the relationship between the total of a 3step stair and the position of it on a 10 x 10 grid.
The three step stair is made up of 6 squares.
X = Stair number or Sn
I will carry out the investigation in the following steps:
 On a 10 x 10 grid move the ‘stairs’ 1 square to the right and find the stair total, by adding all the numbers in the stair together.
 Put the information into a table and look for any pattern or rules.
 Describe any patterns or rules using words and/or algebra.
 Try to explain any patterns found.

Middle
6 80 6
7 86 6
Each time the stair total increases by 6. This takes the previous total to the next total.
To get the formula I multiplied each stair total by 6. I noticed that after each multiplication the only difference between the number and the ST was 44.
Therefore the formula is Dn+(AD)
6n+(506)
6n+44
The stair number is multiplied by 6 and 44 is added to make the stair total.
The formula allows the stair total to be obtained from only the star number.
Part 2
As said in my plan I intend to try different step sizes on the same 10 x 10 grid. I will try a 2step stair.
From my previous investigation I figured that
Conclusion
The total of the difference is 20 + 10 + 11+ 1 +2 = 44
The difference when added together makes the number that must be added at the end of the formula (i.e.6n + 44).
I will see if this applies to all stair sizes.
11  
1  2 
Sn+10  
Sn  Sn+1 
The total of the differences is 11. This is also the end of the rule(3n + 11).
41  
31  23  
21  22  23  
11  12  13  14  
1  2  3  4  5 
Sn+40  
Sn+30  Sn+31  
Sn+20  Sn+21  Sn+22  
Sn+10  Sn+11  Sn+12  Sn+13  
SN  Sn+1  Sn+2  Sn+3  Sn+4 
Total difference = 220
Formula = 15n +220
The rule for finding the differences works.
I will try to find the rules for larger stair numbers.
51  
41  42  
31  32  33  
21  22  23  24  
11  12  13  14  15  
1  2  3  4  5  6 
Sn+50  
Sn+40  Sn+41  
Sn+30  Sn+31  Sn+32  
Sn+20  Sn+21  Sn+22  Sn+23  
Sn+10  Sn+11  Sn+12  Sn+13  Sn+14  
SN  Sn+1  Sn+2  Sn+3  Sn+4  Sn+5 
As there are 21 squares I will say the start of the rules is 21n. The total of the differences is 385. The formula can is 21n + 385.
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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