The polynomial  appearing before the fixed monthly payment c (with ) is a geometric series, which has a simple closed-form expression obtained from observing that  because all but the first and last terms in this difference cancel each other out.
ðə pˌɑːlɪnˈoʊmɪəl ɐpˈɪɹɪŋ bᵻfˌoːɹ ðə fˈɪkst mˈʌnθli pˈeɪmənt sˈiː wɪð ɪz ɐ dʒˌiːoʊmˈɛtɹɪk sˈɪɹiz, wˌɪtʃ hɐz ɐ sˈɪmpəl klˈoʊzdfˈɔːɹm ɛkspɹˈɛʃən əbtˈeɪnd fɹʌm əbzˈɜːvɪŋ ðæt bɪkˈʌz ˈɔːl bˌʌt ðə fˈɜːst ænd lˈæst tˈɜːmz ɪn ðɪs dˈɪfɹəns kˈænsəl ˈiːtʃ ˈʌðɚɹ ˈaʊt.