Let X 1 , X 2 be a random variables with geometric distribution with p,Find...

Step 1
The post contains some elements of a correct approach.
We cannot solve the problem without making some assumptions about the relationship between $X_1$ and $X_2$. We will assume that $X_1$ and $X_2$ are independent.
Let $S=X_1+X_2$. We want to find $Pr(S=n)$. This is 0 if $n\le 1$. So let $n\ge 2$.
Step 2
We can have $S=n$ in several ways. For we could have $X_1=1$ and $X_2=n-1$. Or we could have $X_1=2$ and $X_2=n-2$. And so on up to $X_1=n-1$ and $X_2=1$.
For any k from 1 to $n-1$, we have $Pr(X_1=k\cap X_2=n-k)=(1-p{)}^{k-1}p(1-p{)}^{n-k-1}p=p^2(1-p{)}^{n-2}.$.
Adding up from $k=1$ to $k=n-1$ we find that $Pr(S=n)=(n-1)p^2(1-p{)}^{n-2}.$