I am doing an analysis on the complexity of some set-related algorithm where the input is a random set. One of my setbacks can be formulated as follows:

Pick $k$ distinct numbers out of numbers $[1,n]$ uniformly at random, order them by size and denote them with $\alpha_1, \alpha_2, \dots , \alpha_k$, such that $\alpha_i < \alpha_{i+1}$ for $i\in [1,k-1]$.

I would like to calculate
$$\mathrm{E}\left(\sum_{i=1}^{k}\left(\frac{1}{2}\right)^{\alpha_{i}-i}\right),$$
or at least find a good upperbound to it.