ultimul termen n ·( 2n +1) ·( 3n +1 ) = n ·( 6n² + 5n +1 ) = 6 ·n³ + 5·n² + n
daca n =1 1·3·4 = 6 ·1³ + 5·1² + 1
n =2 2· 5·7 = 6 · 2³ + 5·2² + 2
n =3 3 ·7 ·10 = 6· 3³ + 5 ·3² +2
.....................................................................................
n · ( 2n +1 ) ·( 3n +1 )= 6 ·n³ + 5· n² + n
⇒ suma = 6 · ( 1³ +2³+3³+...+ n³ ) + 5·( 1² +2² +3²+...+n² ) + ( 1 +2+3 +..+ n )
↓ ↓ ↓
[n(n+1) /2]² n( 2n+1)·(n+1) /6 n·( n +1) /2
suma =n·( n +1 ) /2 · [6· n·(n +1) /2 +5· ( 2n +1) /3 + 1]
= n ·( n +1) /2 · [ 3 ·n·( n +1) 5·( 2n+1) /3 + 1 ]
= n ·( n +1) /2 · [ 3n² + 3n + (10n +5) /3 + 1]
= n ·( n +1) /2 · [ ( 9 n² +19n +8 ) /3 ]
