The statement as given is not true. For <em>n</em> = 1, the left side is just the first term, 1, while the right side would be 3(1) - 1 = 2, and 1 ≠ 2.
I think what you meant to write was the equation
1 + 4 + 7 + ... + (3<em>n</em> - 2) = 1/2 <em>n</em> (3<em>n</em> - 1)
which can be easily proved by induction.
For <em>n</em> = 1, we have
1 = 1/2 (1) (3(1) - 1) → 1 = 1
which is of course true.
Now for the inductive step. Assume the equation holds for <em>n</em> = <em>k</em>, so that
1 + 4 + 7 + ... + (3<em>k</em> - 2) = 1/2 <em>k</em> (3<em>k</em> - 1)
and use this to show it holds for <em>n</em> = <em>k</em> + 1.
We would have
1 + 4 + 7 + ... + (3<em>k</em> - 2) + (3(<em>k</em> + 1) - 2)
which reduces to
1/2 <em>k</em> (3<em>k</em> - 1) + (3<em>k</em> + 1)
3/2 <em>k</em>² + 5/2 <em>k</em> + 1
1/2 (3<em>k</em>² + 5<em>k</em> + 2)
1/2 (<em>k</em> + 1) (3<em>k</em> + 2)
which is precisely what the formula gives on the right side for <em>n</em> = <em>k</em> + 1. QED