Questions on writing a Roman-numeral analysis
In bar one, should I write chord I twice or should I write I_________
Also, in bar 3, am I right if I added chord symbols to each of the last three notes or should I add it to only one?
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There's no universal answer for how to write the first measure. Plenty of people (myself included) would just write I and not write anything else until the chord changes, the assumption being that the chord stays the same until we analyze another chord. People will also write I I as you did, and as you said another possibility is I ----. All are correct.
It's very common for chords to change quickly right before the end of a phrase. (In fancy terms, we say that "the harmonic rhythm accelerates as we approach the cadence," but don't worry if you haven't heard those terms yet; you will later!) Furthermore, your chords make complete sense there, so the end of your m. 3 looks great to me!
(Note: People typically use lowercase Roman numerals for minor chords. If that's the case, make sure you write your II as a ii. But if your teacher has asked you to use all uppercase Roman numerals, then stick with II!)
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