= Solution
Put $h=(b-a)/n$, $g_i=g(x_i)$, and $z_0=0$. Under the endpoint-slope convention suggested by the requested connection, the recurrence is $z_{i+1}=z_i+\tfrac h2(g_i+g_{i+1})$. Summing telescopes, so
$$
\boxed{z_n=\frac{b-a}{n}\left[\frac{g_0+g_n}2+\sum_{i=1}^{n-1}g_i\right]=T_n(g).}
$$
Since $G(b)=\int_a^b g(x)\,dx$ by the <fundamental theorem of calculus>, this recurrence gives exactly the <composite trapezoidal rule>. It replaces the average derivative over each strip by the arithmetic mean of its endpoint derivatives.
\b[For the standard <midpoint method>, the literal endpoint-only request is not valid for general $g$.] Both the explicit and implicit one-step midpoint formulas here give instead
$$
\boxed{z_n=h\sum_{i=0}^{n-1}g(x_i+h/2)=M_n(g),}
$$
the <composite midpoint quadrature rule>. It requires half-grid samples. These samples cannot be recovered exactly from $g_0,\ldots,g_n$ for arbitrary smooth $g$: the zero function and $g(x)=\prod_{j=0}^n(x-x_j)^2$ have identical grid values but different midpoint sums. For a smooth $g$, replacing each half-grid sample by $(g_i+g_{i+1})/2$ is an additional approximation, giving $T_n$ with an $O(h^2)$ total difference, rather than an identity. A precise useful relation, obtained by splitting the finer trapezoidal sum into old nodes and midpoints, is
$$
\boxed{T_{2n}(g)=\tfrac12[T_n(g)+M_n(g)],\qquad M_n(g)=2T_{2n}(g)-T_n(g).}
$$
Here $T_{2n}$ uses the finer half-grid, so it does not eliminate the missing data.
If “Mid-Point Method” means the two-step <explicit midpoint linear multistep method>, the recurrence is $z_{i+1}=z_{i-1}+2hg_i$. It gives $z_{2r}=2h\sum_{j=1}^{r}g_{2j-1}$ and $z_{2r+1}=z_1+2h\sum_{j=1}^{r}g_{2j}$, so an odd final index additionally needs a starting value $z_1$. For even $n=2r$, the result is the midpoint quadrature on $r$ double-width strips; equivalently $z_n=2T_n-T_{n/2}$. The question specifies neither even $n$ nor a starter. Thus the trapezoidal expression above is the natural endpoint-average interpretation, while the other common midpoint meanings require the qualifications just given. The ambiguity is present in the original PDF.
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