This question is inspired by the Escalation Battles in Pokémon Shuffle. There's a couple of other Pokémon-related questions on here, but they don't address this specific problem.
The way an Escalation Battle works is, the $n$th time you beat it, you have $n$% chance of catching the Pokemon. If you've already caught the Pokémon, you get items instead. When $n=100,$ you're guaranteed to catch the Pokémon, but the chance of having not caught it by then must be vanishingly small.
I've competed in a few Escalation Battles, and I always seem to catch the Pokémon when $15 \leq n \leq 25.$ It's been years since I studied statistical probability at school, but this doesn't seem very intuitive to me. So I started wondering about the cumulative probability - how likely you are to have caught the Pokémon after $n$ levels.
Is there a general formula to calculate the cumulative probability of having caught the Pokémon after $n$ attempts? How many attempts will it take for the cumulative probability to exceed 50%?