Smallest card sharing a box with card 100

Cards numbered $1$ through $100$ are dealt one by one, and each card is dropped into one of two boxes chosen independently and uniformly at random. Let $X$ be the lowest number in whichever box ends up containing card $100$. Then $\mathbb{E}[X]$ can be expressed as $a(1-a^{-b})$ where $a$ and $b$ are integers. What is $ab$?

$200$. Use $\mathbb{E}[X]=\sum_{k\ge1}\mathbb{P}[X\ge k]$. The event $X\ge k$ means cards $1,\dots,k-1$ all went to the other box, which has probability $2^{-(k-1)}$. Summing for $k=1,\dots,100$ gives $2(1-2^{-100})$, so $a=2$, $b=100$.