HMIC 2016 Problem 5

Let $S=\left\{a_{1}, \ldots, a_{n}\right\}$ be a finite set of positive integers of size $n \geq 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including 1) of distinct numbers in $S$, meaning

$$T=\left\{\sum_{i=1}^{n} a_{i}^{e_{i}} \mid e_{1}, e_{2}, \ldots, e_{n} \geq 0\right\}$$

Show that there is a positive integer $N$ (only depending on $n$ ) such that $T$ contains no arithmetic progression of length $N$.