How to say...

[Shiroisan]

About a month ago I was corrected by someone for saying

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 futotteita

when I apparently should have said

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futottekita

... I'm honestly not entirely sure what the difference is, or where the "ki" came from... Perhaps I simply haven't studied the grammar point yet, but I have no idea how to go about finding it... Can anyone explain this for me?

[SomeCallMeChris]

太っていた would mean to have become and remained fat (or thick) for a period of time in the past.
太ってきた would mean to have gained weight.

Both are valid in some contexts, but the second is more likely in most conversations.

There's a whole lot of times when 行く・来る are used to refer to movement through time, although usually the kanji is not used when it's an auxiliary verb for time. They're similar to but not quite identical to phrases like 'came to be...' and 'went on to... '.

[Shiroisan]

Thanks very much ! thats all i needed.