Prove the equivalence between a CFG and a Context free language
Since this is a very simple "linear" grammar, its language can be determined by looking at the possible derivations.

Clearly a successful derivation has to start with $S\to a^2Tb$, then repeats $k$ times the production $T\to a^3Tb^2$ and closes with $T\to \epsilon$. That will give strings of the form $a^{2+3k}b^{1+2k}$ with $k\ge 0$. (1)

Then we need to establish that these are exactly the strings of the form $a^ib^j$ with $2i=3j+1$. (2)

One direction is easy, strings of the form (1) satisfy (2) -- you solved that yourself -- by observing that 
$2(2+3k) = 3(1+2k)+1$.   

Your proof for the other direction sounds good, but it is not clear to me how one argues that the induction has step size $5$ (other than by trial and error).

The following would work. Start with (2) for natural numbers $i,j$.

This implies $3j+1$ is even (and non-negative).

Or, $3j$ is odd, so $j$ is odd, which means it is of the form $j=2k+1$ for $k\ge 0$.

Then $2i=6k+4$, or $i=3k+2$, and we are done.

This argument seems rather ad hoc. Note the similarity between this exercise and linear algebra. There are two different representations for a line in 2D: using vectors or equations. Here these are:


$\{ (2,1)+ k\cdot(3,2) \mid k\in \mathbb R \}$
$\{ (x,y) \mid 2x-3y=1 \}$


Unfortunately, I do not think linear algebra tricks directly work, as we here must restrict ourselves to integers and not the real numbers.