Pay-per-box hunt for a hidden prize

A prize of $100$ is hidden in one of four closed boxes, each equally likely. You may open boxes one at a time, paying $X$ for each box you open, and you may stop whenever you wish; you keep the prize if you find it. What value of $X$ makes this a fair game?

$X=40$. Opening until you find the prize takes $\tfrac{1+2+3+4}{4}=2.5$ boxes on average, so $100-2.5X=0$ gives $X=40$. Backward induction confirms stopping early never helps at this price: with one, two, three and four boxes left the game is worth $60$, $40$, $20$ and $0$.