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Something i noticed

  • 06-05-2007 01:43AM
    #1
    DanGerMus
    Registered Users, Registered Users 2 Posts: 1,525 ✭✭✭


    This is probably of no use to any one but figured it out while doodlin in maths one day
    1+3=4
    1+3+5=9
    1+3+5+7=16
    1+3+5+7+9=25+11=36+13=49
    Notice pattern all the squares. so it seems summing all the odd numbers from 1 on gives you squares. cool eh. oh and the amount of numbers added is the root.


Comments

  • #2
    rjt
    Registered Users, Registered Users 2 Posts: 219 ✭✭rjt


    happygandalf wrote:
    This is probably of no use to any one but figured it out while doodlin in maths one day
    1+3=4
    1+3+5=9
    1+3+5+7=16
    1+3+5+7+9=25+11=36+13=49
    Notice pattern all the squares. so it seems summing all the odd numbers from 1 on gives you squares. cool eh. oh and the amount of numbers added is the root.

    One reason for this is given by the binomial theorem:

    (a+1)^2 = a^2+2a+1
    And so, if n^2 = 1 + 3 + 5 + ... + 2(n-1)+1
    Thus, (n+1)^2 = n^2+2n+1 = 1 + 3 + 5 + .. + 2(n+1)+1
    So, because 1^2=1, this is true by induction.

    Also, probably a nicer proof, with geometry:
    Think of an NxN square composed of N^2 1x1 bricks. If we want to make this into an (N+1)x(N+1) , we need to add N along the bottom, N along the side and 1 on the bottom right corner (so we need to add 2N+1).
    And a simple induction argument will do the rest.


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