Question: How to write a recursive expression? If I have a sequence $h_t = c a_t + 2c^2 a_{t+1} + 3c^3a_{t+2} + ...$ for some $c<1$, how would I recursively write $h_t$ in terms of $h_{t+1}$ such that $h_t$ should have only $a_t$ term and the rest of $a_i$ for $i>t$ I want to push it to $h_{t+1}$? For example, 
Something like this: $h_t = c a_t + f(h_{t+1})$ is okay but
I don't want something like this: $h_t = \sum_{i=t}^{\infty} c^{i-t+1}a_i + ch_{t+1}$ because $h_t$ has terms of $a_i$ for $i\geq t$. 

I think I have it but you'll have to double check. Let's just write out some things first:
$$
\begin{array}{l*{6}{Cr}}
h_t     &=&ca_t&+&2c^2a_{t+1}&+&3c^3a_{t+2}&+&\cdots\\
h_{t+1} &=&    & &ca_{t+1}   &+&2c^2a_{t+2}&+&3c^3a_{t+3}&+&\cdots\\
h_{t+2} &=&    & &           & &   ca_{t+2}&+&2c^2a_{t+3}&+&3c^3a_{t+4}&+&\cdots
\end{array}
$$
Thus, 
$$h_{t+1}-ch_{t+2} = ca_{t+1} +c^2a_{t+2}+c^3a_{t+3}+\cdots$$
Now notice that 
$$2h_{t+1}-ch_{t+2} = h_{t+1}+(h_{t+1}-ch_{t+2}) = 2ca_{t+1} +3c^2a_{t+2}+4c^3a_{t+3}+\cdots$$
Multiplying by $c$, we get 
$$c(2h_{t+1}-ch_{t+2}) = 2c^2a_{t+1} +3c^3a_{t+2}+4c^4a_{t+3}+\cdots$$
and so 
$$h_t = ca_t + c(2h_{t+1}-ch_{t+2})$$
