# Very Heavy Coin

Calculus Level 5

You are flipping a coin that has a $$\dfrac{n^2-1}{n^2}$$ chance of landing on heads. The value of $$n$$ follows these rules:

• $$n$$ and $$k$$ initially equal 2.

• Whenever the coin lands on heads, $$n$$ increases by 1.

• Whenever the coin lands on tails, $$k$$ increases by one and then $$n$$ is set to equal $$k$$ (e.g. the first time you flip tails, $$k$$ and $$n$$ will both equal 3, then the second time they will equal 4, etc.)

What is the average number of tails you will flip?