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    I'm doing GCSE coursework on steps. I have to find a formula to work out how many matches it takes to build a flight of steps. The sequence goes like this...
    Code:
    x
         x    xx
    x   xx   xxx    etc
    So the match sticks look like this (sorry for the bad drawing :P)

    Thats the second sequence and it uses 28 matches. I've looked at http://www.maths-help.co.uk/Knowldge...h/Question.htm but I don't see how I'd change that formula to make it apply to this problem :confused:
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    write down what the sequence is.... i cant be bothered to check out the site.... although clearly each time u add a flight of steps u add on ur new single block of matchsticks (11 of them without the extra horizontal bit) plus 5n lots of matchsticks where n is the last height of blocks u had.

    so ur T(n+1 th term) = T (n) + 5n + 11
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    Well, the first 3 are:

    12, 28, 49

    When I drew the pictures and worked these out I thought of working out the formula by considering the nth row of the set of stairs.

    So, consider the 3rd row of stairs. By this I mean:

    Code:
    In the instance when n = 3, we have:
    
      #
     ##
    ### <
    
    By the third row, I mean the bottom row in this example with an arrow.
    If you draw a picture in 3d you can see that there are:
    2x4 = 2(3+1) vertical matches
    3x2 + 4 = 3x2 + (3+1) matches on the top layer

    Generalising, we see that in the rth row there are 2(r+1)+2r+(r+1) = 5r+3 matches in the uprights and top layer.

    Sum 1->n (5r+3) = 5n(n+1)/2 +3n

    Then you need to add the number of matches on the bottom layer on the last row which is 3n+1. So the total number of matches is:

    5n(n+1)/2 +3n +3n + 1

    Which simplifies to:

    (5n^2 + 17n +2)/2
 
 
 
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Updated: December 28, 2003

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