A Statement $$\mathrm { S } _ { \mathrm { n } }$$

Essay

A statement
$$\mathrm { S } _ { \mathrm { n } }$$ about the positive integers is given. Write statements
$\mathrm { S } _ { 1 } , \mathrm {~S} _ { 2 }$ and S
$\mathrm { S } _ { 3 }$ , and show that each of these
statements is true.
- $$S _ { n } : 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$$