[Community Question] Statistics: Probability of catching a Pokémon in an Escalation Battle

One of our user asked:

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 Nth time you beat it, you have an 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 ≤ N ≤ 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%?

Calculus: Intersection of hyperplanes in $\mathbb{R}^5$

Consider the $4$ hyperplanes in $\mathbb{R}^5$ given by the equations $$x_1-x_2+x_3+x_4+2x_5=0$$ $$2x_1-x_2+6x_3+2x_4+6x_5=0$$ $$3x_1-3x_2+3x_3+4x_4+7x_5=0$$ $$x_1+x_2+9x_3+x_4+6x_5=0$$ Let $V \leq \mathbb{R}^5$ be the intersection of these hyperplanes. Find a basis for $V$ .

[Community Question] Calculus: calculating 2 constants in a function

One of our user asked: $$M=\{f\in C[0,2\pi],\int_{0}^{2\pi}f(x)sinxdx=\pi,\int_{0}^{2\pi}f(x)sin2xdx=2\pi\} $$ $a,b\in \mathbb R, g\in M, g(x)=asinx+bsin2x,x\in [0,2\pi]$ I've read on the answers that $a=1,b=2$ and I don't know how to calculate them. Can somebody explain me,please? By the way, the problem is to determine $ \int_{0}^{2\pi}(g(x))^2 dx$ so you have to first get the constants $a$ and $b$ .

Calculus: A difficulty in understanding a step in a solution.

Here is the solution: But I could not understand how the last term in the fourth line came from the line before it, could anyone explain this for me please?

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