I will present a bound, but not the solution --not yet! :(

If there are $101$ numbers, then there are $5151$ different pairs. Given that each pair has to yield a different sum, the greatest sum will be, at least, $5151$.

Now, the greatest sum is expected to be $A_n + A_{n+1}$ where $A_n$ is the number asked in the question, and $A_{n+1} < A_n$. Then, the minimum value is when $A_n = 2576$.

So, in general, we can affirm that $A_n \geq 2576$.

I will keep working in a better lower bound, an upper bound and a solution, let's see what comes first.

I will present a bound, but not the solution --not yet! :(

If there are $101$ numbers, then there are $5151$ different pairs. Given that each pair has to yield a different sum, the greatest sum will be, at least, $5151$.

Now, the greatest sum is expected to be $A_n + A_{n+1}$ where $A_n$ is the number asked in the question, and $A_{n+1} < A_n$. Then, the minimum value is when $A_n = 2576$.

So, in general, we can affirm that the solution is greater than $2576$.

I will keep working in a better lower bound, an upper bound and a solution, let's see what comes first.

EDIT: An upper bound, as some answers state is the set $A = \lbrace 2^i \rbrace_i$, which gives $2^{100}$. So we can affirm that the solution is less than $2^{100}$