# Series partial sums

I have a school assignment to do, but I have no idea, where to start. I hope you can help. Here it is:

We have a finite sequence $A = (a_1, a_2,\ldots, a_n)$, length of $A$ is $n$, elements of $A$ are natural numbers.

Let $S(i,j)$ be partial sum of this sequence from $a_i$ to $a_j$, where $1\leq i\leq j\leq n$.

Formally written as $S(i,j) = a_i+a_{i+1}+\cdots+a_j$.

We can call sequence $A$ "nice", when all partial sums $S(i,j)$ are different from each other.

Formally written - For every $1\leq i\leq j\leq n$ and $1\leq k\leq l\leq n$ it holds that "$S(i,j)=S(k,l)$ implies $i=k$ and $j=l$".

E.g. $(1,3,2)$ is "nice" and $(1,4,2,3)$ is not "nice".

Prove that for every natural $n$ a "nice" sequence exists.
What is the smallest possible value of the greatest element of a "nice" sequence $A$ of length $n$? Or in other words - for every $n$ find the smallest natural number $p$ for which a "nice" sequence $A$ of length $n$, in which no element is larger than $p$, exists.
Hint: search for upper and lower bounds on $p$ in the form of appropriate functions of $n$.