Pairwise Distinct Sums

Starting from the smallest element $a_1$ to the largest $a_{10}$ -

Clearly, smallest one i.e. $a_1$ will be 1.

Next, we can have smallest possible $a_2=2$

Thus Further we need to have 8 numbers such that all possible pairs of other 8 must be having their difference more than 1...

Thus $a_3$ can be 3 but $a_4$ can't be 4 as then $4+1=2+3$

Now $a_4$ has to be chosen such that it's difference should be more than $1$ from $a_3$ as $a_2-a_1=1$

So $a_4$ has smallest possible value $5$ due to this condition (because $5-3>1$ )

because now we have $a_3=3$ and $a_1=1$ with difference $2$ , $a_5$ must be greater than $a_4$ by at least 3 ... Hence smallest possible $a_5=a_4+3=8$

Now $a_4-a_1=4$ hence difference of $a_5$ and $a_6$ must be greater than 4....i.e. 5 so $a_6=a_5+5=13$

According to this logic, resulting sequence is the famous FIBONACCI SEQUENCE and we need to find $a_{10}$ where $a_n=a_{n-1}+a_{n-2}$ And $a_1=1,a_2=2$ hence the sequence is

$1,2,3,5,8,13,21,34,55,89$ thus $a_{10}=89$

The problem has the application of fibonacci series. It is a series or set of positive integers such that the sum or the difference of no two pair of numbers is same. S={ 1, 2, 3, 5, 8, 13, 21, 34...} The first and second term of the series is 1 and 2. Each term starting from 3 is the sum of its previous two terms. For Eg. 1+2=3; 2+3=5; 3+5=8; 5+8=13 and so on.. The terms so formed have unique sum and difference pairwise.