= Solution
The state update <matrix> is
$$
A=\begin{pmatrix}1/2&1/2\\1&0\end{pmatrix}.
$$
Its <eigenvalues> are $1$ and $-1/2$, with <eigenvectors> $(1,1)^T$ and $(1,-2)^T$. The initial column $(G_1,G_0)^T=(1,0)^T$ decomposes as $\frac23(1,1)^T+\frac13(1,-2)^T$. Applying $A^i$ and reading the second component gives
$$
\boxed{G_i=\frac23\left[1-\left(-\frac12\right)^i\right]\ \longrightarrow\ \frac23.}
$$
The alternating term decays because its <eigenvalue> has modulus below one. The <convergence of an adjacent-term averaging recurrence> gives the same result by a first-order recurrence for successive differences.
Back to article page
