Start with account containing A.
Buy Γ_0 shares at price X_0.
A_0 = A - Γ_0 X_0.
V_0 = A_0 + Γ_0 X_0 = A - Γ_0 X_0 + Γ_0 X_0 = A.
A_1 = A_0 - Γ_1 X_1 = A - Γ_0 X_0 - Γ_1 X_1.
\begin{aligned} V_1 &= A_1 + (Γ_0 + Γ_1)X_1 \\ &= A - Γ_0 X_0 - Γ_1 X_1 + (Γ_0 + Γ_1)X_1 \\ &= A + Γ_0(X_1 - X_0) \\ \end{aligned}
A_2 = A_1 - Γ_2 X_2 = A - Γ_0 X_0 - Γ_1 X_1 - Γ_2 X_2
\begin{aligned} V_2 &= A_2 + (Γ_0 + Γ_1 + Γ_2)X_2 \\ &= A - Γ_0 X_0 - Γ_1 X_1 - Γ_2 X_2 + (Γ_0 + Γ_1 + Γ_2)X_2 \\ &= A + Γ_0(X_2 - X_0) + Γ_1(X_2 - X_1) \\ \end{aligned}$$
A_n = A - \sum_{j=0}^n Γ_j X_j.
V_n = A + \sum_{j=0}^{n-1} Γ_j(X_n - X_j).
0 = \sum_^n Γ_j, \begin{aligned} A_n &= A - \sum_{j=0}^{n-1} Γ_j X_j + (\sum_{j=0}^{n-1} Γ_j) X_n \\ &= A + \sum_{j=0}^{n-1} (X_n - X_j) \\ \end{aligned}