A sequence of rv $X_{n}$ i.i.d for $n \in \mathbb{N}$ with $S_{0}=0$ and $$S_{n} = \sum_{k=1}^{n}X_{k}$$.
If $k \leq n$ then $$\mathbb{E}[S_{k}|S_{n}] = \frac{k}{n}S_{n}$$
I am thinking that:
$$\mathbb{E}[S_{k}|S_{n}] = \sum_{k=1}^{n} X_{k} \frac{P(X=k)|(P(n-k=n-k)) }{P(n=n)} = \sum_{k=1}^{n} X_{k} \frac{k \cdot 1}{n}= \frac{k}{n}S_{n}$$
By linearity, it suffices to show that $\mathsf{E}[X_i\mid S_n]=S_n/n$ for each $1\le i\le n$. However, the latter equality holds because $X_1,X_2,\ldots,X_n$ are exchangeable r.v.s. (See, e.g., this question for details.)