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 $$h_t = ca_t + c(2h_{t+1}-ch_{t+2})$$