You are right about the base case; 3 and 6 are given, and you've shown that the formula yields them.
But the inductive step is to
assume that
ai=2i+i, not for every
i≥1, which is what you are to
prove, but for every
1≤i≤k for some given k, and use that to prove that it is true for the
next case, namely
ak+1=2k+1+(k+1).
So, what does the definition say
ak+1, using the formula for the two preceding terms? Try to manipulate that to look like
2k+1+(k+1).
No, you don't need to show the value of
a3; you'll be using the definition in the inductive step.