Use mathematical induction to prove the statement is true for all positive integers n.

6 + 12 + 18 + ... + 6n = 3n (n + 1)

First verify the base case by inserting values so you can assume it holds for it n

for n+1 add the next term 6 * (n+1) of the sum to both sides:

6 + 12 + 18 + ... + 6n + 6 (n+1) = 3n (n+1) + 6 (n+1)

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

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

=3 (n+1) ((n+1) + 1)

=term if you had inserted n+1 instead of n, proving correctness