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[Remember Your Squares] Something I Found

  1. Feb 17, 2010 #1
    Let x = 1.
    Let n = Next Odd Number
    Let y = Previous Sum

    x2 = x
    +3 = 4 = (1+x)2
    +5 = 9 = (3+x)2
    +n = n+y = (n-2 + x)2

    You could make a program to list all the squares without invoking the multiplication function or squaring function using a simple loop.

    C++ Example:

    #include "stdafx.h"
    #include <iostream>
    using namespace std;


    int _tmain(int argc, _TCHAR* argv[])
    {
    cout<<"Squares: \n\n";
    int sum=1,nextOdd=1; // Sum is Starting Integer Squared, Declared X in the For Loop below

    for(int x=1;x<100;x++)
    {
    cout<<x<<" Squared is "<<sum<<"\n";
    nextOdd+=2;
    sum+=nextOdd;
    }
    system("pause");
    }
     
  2. jcsd
  3. Feb 17, 2010 #2

    CRGreathouse

    User Avatar
    Science Advisor
    Homework Helper

    You've discovered that
    [tex]\sum_{k=1}^n2k-1=n^2[/tex].
    Congratulations.

    Can you transform that into a formula for [tex]\sum_{k=1}^nk[/tex]? Can you find one for [tex]\sum_{k=1}^nk^2[/tex]?

    You can check your work afterward, and even glimpse what's beyond:
    http://mathworld.wolfram.com/FaulhabersFormula.html
     
  4. Feb 17, 2010 #3
    Are you being sarcastic with "congratulations"?
     
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