Sum of n terms 1x2+2x3+3x4+4x5 + ...

1/3 (n+1) ³

Step-by-step explanation:

1x2+2x3+3x4+4x5 + ... = 1²+1+2²+2+3²+3 + ... + n²+n+1=

= (1²+2²+3² + ... + n²) + (1+2+3 + ... + n+1) =

=1/6n (n+1) (2n+1) + 1/2 (n+1) (1+n+1) =

=1/6 (n+1) (n (2n+1) + 3 (n+2)) =

=1/6 (n+1) (2n²+4n+2) =

=1/6 (n+1) * 2 (n+1) ²=

=1/3 (n+1) ³