Assume $1\leq\ k<m<n$ are positive integers and $X_1,X_2,...X_n$ are i.i.d. Geometric($p$) random variables. For all $j\geq\ k$ define $I_j=[(i_1,i_2,...,i_k):1\leq\ i_1<i_2<...<i_k\leq\ j]$. Define $S_j=\sum\ _{(i_1,i_2,...,i_k)\in\ I_j}ln(2+(X_{i_1})^3+(X_{i_2})^3+...+(X_{i_k})^3)$ Show that $S_m/S_n$ has finite expectation and compute this expectation.
So this is the question. I cannot figure out why this question needed to be so artificial, not to add that I am facing difficulty in solving this. Please treat this as a homework question and give me some hints and not the complete solution.