The い in 住んでいます makes it continuous. So, 住んでいました is a past continuous word and can be translated as "~was living~". On the other hand, 住みます is not continuous, hence the meaning of the word 住みました is...yep, you guessed it, "~(have) lived~" !!! So, neither you nor your book is wrong, both are correct. Here is how the English translation of the 2 sentences should be like=> 1. "アメリカに住んでいました。" "Amerika ni sunde imashita." ==> I was living in America. 2. "アメリカに住みました。" "Amerika ni sumimashita。" ==> I (have) lived in America. ***[I don't know if you've already learnt the rule of 'て-Form' so I will mention one thing here just in case it helps you. According to the dictionary form, verbs ending with ～む (in this case, 住む, 'sumu') changes to ～んで (住んで, 'sunde') when changing to other tenses etc.] Hope my answer was helpful, even though I am not a native Japanese speaker! Take care and enjoy learning Japanese... \(^__<)/

住んでいるmeans you have done the action for a long time and maybe you can translate it into `used to have been living in some place` and 住みました is `once you have lived in some place` →It's my option and the native speaker will get a better answer~