Compute variance of a power Poisson process

Question

Let $N_t$ be a Poisson process with parameter $\lambda$. Let $X_t = (1+a)^{N_t}$. Compute $Var((1+a)^{N_t})$.

My attempt:

First I tried to compute the MGF, but I could not simplify it.

Then I try to compute it by definition:

$Var(X_t) = E[(1+a)^{2N_t}] - (E[(1+a)^{N_t}])^2$

I can easily compute: $E[(1+a)^{N_t}] = e^{\lambda t a}$

But I cannot simplify the quantity:

$$E[(1+a)^{2N_t}] = \sum_{k=0}^{\infty} (1+a)^{2k} \dfrac{(\lambda t)^k e^{-\lambda t}}{k!} =e^{-\lambda t} \sum_{k=0}^{\infty} (1+a)^{2k} \dfrac{(\lambda t)^k }{k!}$$

How can I compute this sum? There must be some clever way!

Answer 1

$$\sum_{k=0}^{\infty}(1+a)^{2k}\frac{(λt)^k}{k!}=\sum_{k=0}^{\infty}\frac{\left((1+a)^2λt\right)^k}{k!}=e^{(1+a)^2λt}$$