The expected value of the smallest number in sample $S$ is:

Question

We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest number in sample $S$ is:

1. $\frac{1}{n}$
2. $2$
3. $ \sqrt{n}$
4. $ n$

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My attempt :

I assume $∅$ was excluded in $S$. Then,

$$ E = \frac 1{2^1}\times(2^1) + \frac 1{2^2}\times(2^2) + \cdots \frac 1{2^n}\times(2^n)= \overbrace {1+\cdots +1}^{n\;terms}=n$$

Can you explain in formal/alternative way, please?