The following polynomial defines the interactions of two players in a Game Theory problem in Ginti's H. "Game Theory Evolving", where $p \in (0,1), v \in \Bbb R, k \in \Bbb Z$:
\begin{eqnarray*} v_1 & = & (v-1)p + (v-3)p(1-p)^2 + ... + (v-k+2)p(1-p)^{k-3}\;,\\ v_2 & = & p(1-p) + 3p(1-p)^3 + ... +(k-2)p(1-p)^{k-2}\;. \end{eqnarray*}
The 'Answers to problems' section of the book hints of the following for the required $v$:
$$v = v_1 - v_2 -(k-2)(1-p)^{k-1}$$
and states that after simplifications we should arrive at
$$v = \frac {p^3 - p^2(v+3) + 4p + (vp - 2)(1 - (1 - p)^k)} {(2-p)(1-p)p}\;.$$
Please help on how to arrive at such simplification. So far I've rewritten the polynomial in closed form but lost the dependency on $k$, and does not land at the stated result. To express the respective closed forms in terms of $k $ is a must, it seems, since the problem then proceeds to derive $v$ with respect to $k$, which yields
$$v_k = \frac {(1 - p)^{k - 1} (p v - 2) \log(1 - p)} {(p - 2) p}\;.$$
Hints most appreciated here.