Examine Kari’s proof by induction below.

I know $3+6+9+\dots+3n=\frac{3}{2}n\left(n+1\right)$ is true for $n = 2$ because $3+6=\frac{3}{2}\left(2\right)\left(2+1\right)=9$. If I assume that $3+6+9+\dots+3k=\frac{3}{2}k\left(k+1\right)$ is true, then:

Therefore, the relationship is true for $n = k + 1$ whenever it is true for $n = k$.

1. What did she prove?

   Hint (a):

   What is the relationship? When (for what numbers) is it true?

2. How could she adjust her proof so that she can know for sure that the relationship is true for all $n\ge1$?

   Hint (b):

   Which value of $n$ should she check instead of the one she did?