Squared triangular number

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the sum of the first n <a href="/facts/Cube_(algebra)/SqmlEEog">cubes</a> is the <a href="/facts/Square_number/jfakASPK">square</a> of the nth <a href="/facts/Triangular_number/KHonHtQJ">triangular number</a>. That is,

1
          
            3
          
        
        +
        
          2
          
            3
          
        
        +
        
          3
          
            3
          
        
        +
        ⋯
        +
        
          n
          
            3
          
        
        =
        
          
            (
            
              1
              +
              2
              +
              3
              +
              ⋯
              +
              n
            
            )
          
          
            2
          
        
        .
      
    
    {\displaystyle 1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(1+2+3+\cdots +n\right)^{2}.}

The same equation may be written more compactly using the mathematical notation for <a href="/facts/Summation/V2WUxoKn">summation</a>:

∑
          
            k
            =
            1
          
          
            n
          
        
        
          k
          
            3
          
        
        =
        
          
            (
            
              
                ∑
                
                  k
                  =
                  1
                
                
                  n
                
              
              k
            
            )
          
          
            2
          
        
        .
      
    
    {\displaystyle \sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.}

This <a href="/facts/Identity_(mathematics)/LR46j4uR">identity</a> is sometimes called Nicomachus's theorem, after <a href="/facts/Nicomachus/EdlB4L3G">Nicomachus of Gerasa</a> (c. 60 – c. 120 CE).

Squared triangular number open-in-new

Squared triangular number