I lost interest after I had been a couple of pages in.
Can we use this sentence in formal situations?
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The simple past tense would sound better:
I lost interest after I was a couple of pages in.
Can this be used in a formal situation?
Well... the above mentioned sentence is more like typical spoken English rather than formal written English.
To be clear - it sounds informal. But not in some extraordinary way. It's not slang, for example.
Often, written essays and articles take an informal tone. In those cases, you might be able to use this phrasing. Other times, where a truly formal style is required, you could rewrite the sentence. One example: "The author is not able to hold the reader's interest, droning on about trivial details which have little relevance to the main topic."