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\text{G'}(t) = (1-q^n)\left[\dfrac{tq^n}{(1-q^nt}^2} + \dfrac{1}{1-q^nt} \right] \text{ and } E(R) = G(1) = \dfrac{q^n}{1-q^n} + 1
\\ a_k = 3a_{k-1} - a_{k-2} \\[br]\\ \implies a_k = 3a_{k-1} - \frac{1+a_{k-1}^2}{a_k} \\[br]\\ \implies a_k^2 = 3a_k a_{k-1} - 1 - a_{k-1}^2 \\[br]\\ \implies a_k^2 + 1 = 3a_k a_{k-1} - a_{k-1}^2 \\[br]\\ \implies a_{k+1}a_{k-1} = 3a_k a_{k-1} - a_{k-1}^2 \\ [br]\\ \implies a_{k+1} = 3a_k - a_{k-1}
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