Geometric progressions

Algebra Level 4

99 infinitely long lines of people approach a pile of 1000000 cakes.

In the first line of people, the first person eats \(\frac{1}{3}\) of the cakes, the second person eats \(\frac{1}{3}\) of what the first person of the first line ate, the third person eats \(\frac{1}{3}\) of what the second person ate, and so on. After the first line has passed, the first person of the second line eats \(\frac{1}{4}\) of the remaining cakes, and the second person eats \(\frac{1}{4}\) of what the first person of the second line ate, and so on. After all the lines have passed, a greedy person eats 7982 cakes. How many cakes remain?

Note: people are allowed to eat fractions of cakes.

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