# Number Stairs Part 1: To investigate the relationship between the stair total and the position of the stair shape on the grid, for ot

Mathematics GCSE Coursework: Number Stairs                4/28/2007

Mathematics GCSE Coursework: Number Stairs

Part 1: To investigate the relationship between the stair total and the position of the stair shape on the grid, for other 3-step stairs. The number in the bottom left corner of the stair shape labels the position of the stair shape. The aim of this part of the investigation is to find a formula to work out the s-total (stair total) by knowing the s-number (stair number).

25 + 26 + 27 + 35 + 36 + 45 = 194

s-number = 25

s-total = 194

1 + 2 + 3 + 11 + 12 + 21 = 50

s-number = 1

s-total = 50

2 + 3 + 4 + 12 + 13 + 22 = 56

s-number = 2

s-total = 56

3 + 4 + 5 + 13 + 14 + 23 = 62

s-number = 3

s-total = 62

4 + 5 + 6 + 14 + 15 + 24 = 68

s-number = 4

s-total = 68

Another observation here is that in each column of the stair shape, the difference between these rows of numbers in the same column, is 10. (Apart from the last column, which only consists of 1 number.) The difference between these numbers is 10 because in the grid, each row contains 10 numbers.

Table of Results:

To go from one term to the next, simply add 6 each time. However, this does not show the relationship between the s-number and the s-total. This sequence (adding/subtracting a number each time), is known as an arithmetic sequence. A formula to find out the relationship between the s-number and the s-total is: first term + common difference x ( n – 1 ). The first term here is 50, and the common difference is 6.

50 + 6(n – 1)

=        50 + 6n – 6

=        6n + 44

5 + 6 + 7 + 15 + 16 + 25 = 74

(6 x 5) + 44 = 74

s-number = 5

s-total = 74

6 + 7 + 8 + 16 + 17 + 26 = 80

(6 x 6) + 44 = 80

s-number = 6

s-total = 80

26 + 27 + 28 + 36 + 37 + 46 = 200

(6 x 26) + 44 = 200

s-number = 26

s-total = 200

After testing the formula to see if it works, a proof must be presented to show that it works for any s-number (with a few exceptions – see “Conclusion”). If the s-number is represented as ‘n’:

As mentioned in the aim, the bottom ...