圖解連續正整數的立方和

參閱方格紙上的圖形可知
13 + 23 = 12+2( 1 × 2 )+22 = ( 1+2 )2
( 13 + 23  ) +33 = ( 1 + 2 )2+2[ ( 1 + 2 ) × 3 ] +32 = ( 1 + 2 + 3 )2

( 13 + 23 + 33  ) + 43 = ( 1 + 2 + 3 )2 + 2[ ( 1 + 2 + 3 ) × 4 ] + 42 = ( 1 + 2 + 3 + 4 )2

( 13 + 23 + 33 + 43  ) + 53 = ( 1 + 2 + 3 +4 )2 + 2[ ( 1 + 2 + 3 + 4 ) × 5 ] + 52 = ( 1 + 2 + 3 + 4 + 5 )2

( 13 + 23 + 33 + 43 + 53  ) + 63 = ( 1 + 2 + 3 + 4 +5 )2 + 2[ ( 1 + 2 + 3 + 4 + 5 ) × 6 ] + 62 = ( 1 + 2 + 3 + 4 + 5 + 6 )2

推廣

( 13 + 23 + 33 +••••••+ (n-1)3  ) + n3 = ( 1 + 2 + 3 +••••••+ (n-1) )2 + 2[ ( 1 + 2 + 3 ++••••••+ (n-1) ) × n ] + n2 = ( 1+2+3+••••••+n )2=$(\frac{n(n+1)}{2})^2$


n =         

$\small\displaystyle\sum_{k=1}^{n}{k^3}=(\frac{n(n+1)}{2})^2=$             

 

相關連結：圖解連續正整數平方和公式



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