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At the end of each turn, Toxic deals more damage. An example is in Showdown. Click on your poisoned Pokemon and it will show, let's say, 31 damage of Toxic will be dealt. After that turn, click on your Pokemon again and it will say that Toxic will do 38 damage (This is just an example.) I know that the damage continually increases, but by how much? Is it by a certain number. Is it slightly random? How much more damage does it deal?

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Oh I found a question related to this!

https://pokemondb.net/pokebase/188371/what-stat-does-poison-decrease

This also mentions about damage done by toxic at the end of each turn.

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Toxic badly poisons the target, and has an accuracy of 85%. When the target takes damage from the poison, the damage done will be N x, where N starts at 1 and x is 1/16 of the target's maximum HP (rounded down, but not less than 1). While a Pokémon is badly poisoned, N increases by 1 each time it takes either Toxic or Leech Seed damage (meaning it may increase twice per turn if a Pokémon is affected by both). When an afflicted Pokémon is affected by Haze, switches out, or when the battle ends, the Pokémon's poison status becomes regular poison.
If a badly poisoned Pokémon successfully uses Rest, it will be cured of poison, but N is not reset; if it then suffers burn, Leech Seed or poison damage, that damage will still be calculated as N
x, and the N value will still increase by 1 each time (however, if the Pokémon is poisoned with Toxic, the N value will be reset to 1).
Toxic cannot affect Poison-type Pokemon.
In Pokémon Stadium, the N value of Toxic is reset by Rest.

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It means, N=1 and starts as 1 and increases by 1 every turn. The damage is calculated by N multiplied by 1/16 of the Pokemon's HP. Since N increases by 1 each turn, second turn the Pokemon will take N = 2 multiplied by 1/16 of the Pokemon's maximum HP.
Example: A Tyranitar with 252 HP, 31 IVs at Level 100 has 404 HP. When it gets badly poisoned, first turn, it will be N multiplied by 1/16 of its HP, which is 1 multiply by 25.25, rounded down 25 HP Points, which is rounded down 6% of its HP. Next turn, N = 2, then it will be 2 * 25 = 50, which is 12% rounded on, and so on. Third turn: it's 3 multiply by 25 = 75, which is rounded down 18%. Fourth turn: 4 multiply by 25 = 100, rounded down 24%. Fifth turn: 5 multiply by 25 = 125, rounded down 30%. Hope this helps!

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I don't understand...
Ahem. "When the target takes damage from the poison, the damage done will be N x, where N starts at 1 and x is 1/16 of the target's maximum HP (rounded down, but not less than 1). While a Pokémon is badly poisoned, N increases by 1 each time it takes either Toxic." I mean, first turn of being badly poisoned, it takes N=1 multiplied by 1/16 of Pokemon's HP. The N counter increases by 1 every turn, meaning second turn would be 2*1/16, and so an.
Thanks. I understand now.
I think I should calculate for at least 4 turns more, so that I understand more clearly. Happy to help!
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Toxic badly poisons the target, and has an accuracy of 85%. When the target takes damage from the poison, the damage done will be N x, where N starts at 1 and x is 1/16 of the target's maximum HP (rounded down, but not less than 1).

It means that at first the damage dealt would be 1/16 of the total HP, then it is 2/16 (i.e. 1/8), then 3/16 and so on.

It says that N starts at 1, which means it will be 2, then 3 and so on.

Source

Let me give you an example. Let someone's HP stat be 496. At the first turn, the opponent outspeeds you and poison you. You would lose 496 x 1/16 which is equal to 31 HP. Remaining would be 465 HP.
Now assume you get hit by your opponent and lose 161 HP. This time you lose 2/16 of the remaining HP which is 1/8 of (463 - 161) = 1/8 of 304 = 48 HP.
This process continues till you faint or get healed.

(The numbers are mentioned in the question)

Hope this helps :)

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Well, its right both the way you tell and I tell.
You helped too. Thanks