The stair-total for this stair shape is 24 + 25 + 26 + 34 + 35 + 44 = 188
The stair-total for this stair shape is 26 +27 + 28 + 36 + 37 + 46 = 200
The stair-total for this stair shape is 27 + 28+ 29 + 37 + 38 + 47= 206
The stair-total for this stair shape is 3 + 4 + 5 + 13 + 14 + 23 = 62
The stair-total for this stair shape is 4 + 5 + 6 + 14 + 15 + 23 = 68
This table summarizes these results :
In order to find a formula which give the stair total when I am given the stair number,
I am going to put the stair number as the position and the stair total as the term for the
sequence:
I have noticed that there is an increase of 6 between two consecutive terms in this
arithmetic sequence. Therefore the term rule must be 6 x Position ± number .
We can notice that the term is always 6 times the position, Add 44 .
So, for the bottom stair of any 3-stair shape on a 10 x 10 grid, the formula must be
Tn = 6n + 44, where 'n' is the stair number, and Tn is the stair total.
I have realised that the 6n in the formula must come from the number of squares in the
stair shape.
I have represented 'n' as the stair number, and the other numbers in relation to the stair number 'n' .
Stair total = n + (n + 1) + (n + 2) + (n + 10) + (n + 11) + (n + 20)
= n + n + 1 + n + 2 + n + 10 + n + 11 + n + 20
=6n + 44
This means that no matter where I translate this stair shape around this grid, the values in terms of ‘n’ will always be the same.
To check if this formula works for other stair numbers, I will try another numbe 55
Stair-total = 6n + 44
( using the formula ) = 6(55) + 44
= 330 + 44
= 374
Stair-total = 55 + 56 + 57 + 65 + 66 + 75
( by adding ) = 374 ✓
This shows that my formula must work for all stair-numbers in 3-stair shapes on the 10 x 10 grid.
However, I have observed that you could just sub any number as 'n' into the formula and
you could still get a stair total even if that stair shape cannot actually be drawn on the grid.
For instance, we could substitute 'n' by 10 into the formula like this:
Tn = 6n + 44
= 6(10) + 44
= 60 + 44
= 104
However it is impossible to draw a stair shape with the bottom left-hand number as
10, simply because it would not fit.
