How do you read this context-free language?

say you want to make a Pushdown Automaton to recognize this language.

What exactly does the +1 mean? I see in the example it just pushes an a to the stack before arriving at an acceptance state but I don't quite get why.

• What +1 are you referring to? The only +1 I see is in the definition of the language on the top of the image. In that, it means the same as $3j+1$ means anywhere else in mathematics. The "+1" in the definition of the language doesn't "push" anything. I can't tell what your question is. Are you familiar and comfortable with set notation? If not, maybe you should ask about that on Mathematics.SE. – D.W. Nov 26 '16 at 23:37
• Hey, yes thats what I was talking about. I don't get what am I supposed to do with the +1 – crystyxn Nov 27 '16 at 12:22

The language defined as $\{a^ib^j\mid 2i=3j+1\}$ is just the set of all strings of $i$ copies of $a$ followed by $j$ copies of $b$, where $2i=3j+1$. Trying various values of $i$ and $j$ and realizing that both $i$ and $j$ must be integers, we see that the admissible solutions are $$(i,j)=(2, 1), (5, 3), (8, 5), (11, 7), \dotsc$$ For example, with $i=5, j=3$ we see that $2i=2(5) =10=3(3)+1=3j+1$. The corresponding strings are $$aab,\quad a^5b^3,\quad a^8b^5,\quad a^{11}b^7 \dotsc$$ and so on.
With a bit of thought, we can make a PDA for this language by pushing two $a$ markers on the stack for each $a$ we read and then popping off three $a$s for each $b$ we see. If we wind up with a single $a$ on the stack after having read all the $b$s, we pop that off and go to the accept state.
It's worth noting that the PDA you picture does not accept the language described. That one accepts the language $\{a^nb^{2n-1}\mid n>0\}$.