Let $X_1$ and $X_2$ geometric random variables independent with parameters $p_1$ and $p_2$. Find the joint distribution of $(X_1,X_1 + X_2)$.
My attempt is the following:
\begin{align*} P[X_1=x_1,X_1 + X_2=x_2] &= P[X_1 + X_2=x_2|X_1=x_1]P[X_1=x_1]\\ &= P[x_1 + X_2=x_2] p_1(1-p_1)^{x_1}\\ &= P[X_2=x_2-x_1] p_1(1-p_1)^{x_1}\\ &= p_2(1-p_2)^{x_2-x_1} p_1(1-p_1)^{x_1}\\ \end{align*}
Is this right? I do not know because I did not use the independence of $X_1$ and $X_2$.
You actually did use independence when you wrote $$P[X_1 + X_2=x_2|X_1=x_1] =P[x_1 + X_2=x_2|X_1=x_1]=P[X_2=x_2-x_1].$$ Note that the equality you wrote is for $x_2\geqslant x_1$, for $x_2<x_1$ is $0$.