Sorry, I wanted to ask this on the finance/money forum, but they don't support LaTeX there.
Let's say we are valuing a company using the DCF methodology with a 5-year projection period.
We project free cash flows of $F_{1},\ldots,F_{5}$. Then if $w$ is the WACC of this company and $g$ is the perpetual growth rate from year 5 forward, the sum of the future cash flows discounted at $w$ is
$$V_{1}:=F_{1}(1+w)^{-1}+\ldots+F_{5}(1+w)^{-5}+F_{5}\sum_{t=6}^{\infty}\frac{(1+g)^{t-5}}{(1+w)^{t}}.$$
This formula for the Gordon Growth model replaces the infinite sum with the easily computed geometric series $$F_{5}\sum_{t=1}^{\infty}\frac{(1+g)^{t}}{(1+w)^{t}}=F_{5}\frac{1+g}{w-g},$$ and therefore (basically) DOUBLE COUNTS (!!) the cash flows $F_{1},\ldots,F_{5}$ to get $$\begin{align*} V_{2}&:=F_{1}(1+w)^{-1}+\ldots+F_{5}(1+w)^{-5}+F_{5}\sum_{t=1}^{\infty}\frac{(1+g)^{t}}{(1+w)^{t}}\\ &=F_{1}(1+w)^{-1}+\ldots+F_{5}(1+w)^{-5}+F_{5}\frac{1+g}{w-g}\\ &\gg V_{1}.\end{align*}$$
What am I missing here?
EDIT
Even if you could convince me of the legitimacy of $F_{5}(1+g)^{t-5}\mapsto F_{1}(1+g)^{t}$ in order to get a uniformly indexed sum (and hence a geometric series), i.e. $F_{5}$ equals the 6-fold growth of $F_{0}$ before we first start to sum it, I would still be very hard to convince of the legitimacy that we should also not truncate the series and re-index the sum at $t=1$.