Financial Arithmetic Q1

At the beginning of January 2025, James took a housing loan of \(\$550\;{000}\) from National Housing Board (NHB) that charges \(2.60\%\) at the end of every year. James pays back \(\$3900\) in the middle of every month, starting from January 2025.

1. Show that the amount that James owes NHB at the start of \(n\) year is \(1,846,800 - 1,296,800(1.026)^{n-1}\).
2. Determine the earliest month and year he will fully repay his loan and the amount of his final instalment.
3. Calculate the total interest he will have paid upon completing the loan repayment.