# Number stairsMy aim is to investigate the relationship between the stair total and the position of the stair shape on the grid for 3 step

Jaie Wilde

Number stairs

My aim is to investigate the relationship between the stair total and the position of the stair shape on the grid for 3 step stairs and to investigate further the relationship between the stair totals and other step stairs on other number grids.

To begin I will try to identify a pattern. Taking my example I will take all the three stair steps on the bottom row and see if there are any similarities between the results.

1. 1+2+3+11+12+21=50
2. 2+3+4+12+13+22=56
3. 3+4+5+13+14+23=62
4. 4+5+6+14+15+24=68
5. 5+6+7+15+16+25=74
6. 6+7+8+16+17+26=80
7. 7+8+9+17+18+27=86
8. 8+9+10+18+19+28=92

Therefore I can now find a formula. The general term for an arithmetic sequence is Un=ab+c. The terms go up in sixes and this tells me that the nth term will include 6 lots of n or 6n.  For the first term n=1, so 6n=6. But the first term is 50 which is 44 more than 6n. This suggests that the formula is 6n+44.

Trying a few values of n will help prove that my formula is correct.

(6 multiplied by 1) +44 =50.

(6 multiplied by 2) +44 =56.

(6 multiplied by 3) +44 =62.

(6 multiplied by 4) +44 =68.

(6 multiplied by 5) +44 =72.

I will now pick a random 3 step stair and test my formula.

Step 1) Locate n which is in bottom left corner.

N=56

Step 2) Insert n into formula 6n+44.

Step3) (6 multiplied by 56) +44= 380 which equals the sum of the numbers.

The formula has worked.

With this formula I can now identify the sum of any 3 step stairs. I will now go on to try and identify similar formulas for bigger stair steps.

4 step stairs

I will begin by using the same method as I used for finding the formula for 3 step stairs by taking all the 4 step stairs on the bottom row and finding the sum.

1)1+2+3+4+11+12+13+21+22+31=120

2)2+3+4+5+12+13+14+22+23+32=130

3)3+4+5+6+13+14+15+23+24+33=140

4)4+5+6+7+14+15+16+24+25+34=150

5)5+6+7+8+15+16+17+25+26+35=160

6)6+7+8+9+16+17+18+26+27+36=170

7)7+8+9+10+17+18+19+27+28+37=180

Using the formula Un=ab+c I can tell that the terms go up in tens. This tells me that the nth term will include 10 lots of n or 10n.  For the first term n=1, so 6n=6. But the first term is 120 which is 110 more than 10n. This suggests that the formula is 10N+110.

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