Prove the following statement:
Let $X_{1},X_{2}, \dots ,X_{n}$ be a set of exchangeable random variables. Then, $$E\left(\frac{X_{1}+X_{2}+X_{3}+\dots +X_{k}}{X_{1}+X_{2}+X_{3}+\dots +X_{n}}\right) = \frac{k}{n} , \qquad 1 \leq k \leq n. $$
I tried writing $\frac{X_{1}+X_{2}+X_{3}+...+X_{k}}{X_{1}+X_{2}+X_{3}+...+X_{n}}$ in a nice form so that I can make calculations easier but only i could try is $1 - \frac{X_{k+1}+X_{k+2}+X_{k+3}+...+X_{n}}{X_{1}+X_{2}+X_{3}+...+X_{n}}$ and I am facing the same looking sum again,also can we use some trick as we know $E(X_{1}) = E(X_{i})$ for $i = 1,2,3,...,n$?
Also induction may help,i think.
Source: Problem 4.3-6 p. 126 of An introduction to Probability and Statistics by Rohatgi and Saleh.
Any help is great!