Let ${e_t}$ be a zero mean white noise process with variance $\sigma^2$, and let c be a constant with $|c|<1$. Define $Y_t=cY_{t-1} + e_t$ with $Y_1 = e_1$
1) Show that E[$Y_t$] = 0
2) Show that $\operatorname{ Var} [Y_t]= \sigma^2(1+c^2+c^4+c^6+...+c^{2t-2}$)
What I have so far
The series expands as follows??
$Y_1 = e_1$
$Y_2 = cY_1+e_2=ce_1+e_2$
$Y_3 = cY_2+e_3=c(ce_1+e_2)+e_3$
$Y_4 = cY_3+e_4=c(c^2e_1+ce_2+e_3)+e_4$
$Y_n$ =c($c^{n-2}$$e_1$+$c^{n-3}$$e_2$+$c$$e_{n-1}$)+$e_n$
for part 1) here's what I have
E[$Y_t$]=E[$Y_t=cY_{t-1} + e_t$]=E[$cY_{t-1}$] + E[$e_t$] = c*0+0=0
for 2) $\operatorname{ Var}[Y_t] = \operatorname{ Var}[cY_{t-1} + e_t] = \operatorname{ Var}[cY_{t-1}] + \operatorname{ Var}[e_t] = c^2\operatorname{ Var}[Y_{t-1}]+\sigma^2$
I don't really know how to turn this into the above answer. I see why the the $\sigma^2$ is a common term, but not where the $2t-2$ comes from. Any help would be great.
The step $\mathrm{Var}\left[cY_{t-1}+e_t\right]= \mathrm{Var}\left[cY_{t-1}\right]+\mathrm{Var}\left[e_t\right]$ needs to be justified, for example using the fact that $Y_{t-1}$ is a linear combination of $e_i$'s for $1\leqslant i\leqslant t-1$. Defining $a_t:=\mathrm{Var}\left[Y_{t}\right]$, we have the recurrence relation $a_t=c^2a_{t-1}+\sigma^2$ and $a_1=\sigma^2$. We then conclude by induction.