Advertisement
If you have a new account but are having problems posting or verifying your account, please email us on hello@boards.ie for help. Thanks :)
Hello all! Please ensure that you are posting a new thread or question in the appropriate forum. The Feedback forum is overwhelmed with questions that are having to be moved elsewhere. If you need help to verify your account contact hello@boards.ie

Something i noticed

  • 06-05-2007 01:43AM
    #1
    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

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


    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.


Advertisement