# Multiperiod return formulae with dividends

I have a question about returns when dividends are 'paid'. Firstly, will write down some definitions:

Let $P_t$ be the price of an asset at time t. Assuming no dividends the net return over the holding period from time $t-1$ to time $t$ is \begin{equation} R_t = \dfrac{P_t- P_{t-1}}{P_{t-1}} \end{equation}

The gross return is defined as $R_t + 1$.

The gross return over the most recent $k$ periods is the the product of the single period gross returns (from time $t-k$ to time $t$) \begin{equation} 1+ R_t(k) = \dfrac{P_t}{P_{t-k}} = \Big(\dfrac{P_t}{P_{t-1}}\Big)\Big(\dfrac{P_{t-1}}{P_{t-2}}\Big)\cdots \Big(\dfrac{P_{t-k+1}}{P_{t-k}}\Big) \end{equation}

However, adjusting for dividends. if a dividend $D_t$ is paid prior to time $t$,then the gross return at time $t$ is defined as \begin{equation}1 + R_t = \dfrac{P_t+ D_t}{P_{t-1}} \end{equation}

Here is what I don't understand: Multiple period gross returns are products of single period gross returns so that: \begin{equation} 1+ R_t(k)= \Big(\dfrac{P_t + D_t}{P_{t-1}}\Big)\Big(\dfrac{P_{t-1}+ D_{t-1}}{P_{t-2}}\Big)\cdots \Big(\dfrac{P_{t-k+1} + P_{t-k+1} }{P_{t-k}}\Big) \end{equation}

For the last formula(with dividends) $1 + R_t(k) \neq \dfrac{P_{t} + D_t}{P_{t-k}}$, as far as I can see, so what is it then?