QUESTION::
Let $X_1$, $X_2$ , ... $X_n$ be IID random variables with
P($X_i$ = 1) = P($X_i$ = -1) = $p$
and P($X_i$ = 0) = $1- 2p$ $\forall i = 1, ... , n$
Define $a_n$ = P ($\prod_{i=1}^n X_i = 1$) ; $b_n$ = P ($\prod_{i=1}^n X_i = -1$) ; $c_n$ = P ($\prod_{i=1}^n X_i = 0$)
Estimate convergence of $ a_n , b_n , c_n $ as $ n \to \infty$
SOLUTION :: $a_n = p^n = b_n $ and $c_n = (1-2p)^n$
As $ n \to \infty$
$ a_n \to 0 , b_n \to 0 , c_n \to 1$
Is this the right approach ? Or is there something missing ?
Hints:
$$c_n=1-P\left(\prod_{i=1}^nX_i\neq0\right)=1-(2p)^n$$
As Kavi made clear in his comments also your calculations of $a_n$ and $b_n$ are not correct.
Further note that $a_n+b_n+c_n=1$ for every $n$.
Try to prove that $a_n=b_n$ for every $n$ which leads to $a_n=b_n=\frac12(1-c_n)$.