Yvette opens a bank account and deposits \(\$x\) on 1 January 2025. On the first day of each subsequent month, she deposits \(\$y\) more than in the previous month, such that she deposits \(\$(x+y)\) on 1 February 2025, \(\$(x+2y)\) on 1 March 2025, and so on. Given that Yvette deposits \(\$825\) on 1 October 2025 and she will have \(\$24\,000\) in her account on 1 December 2026, find the values of \(x\) and \(y\).[3]
Ivan obtained a bank loan of \(\$24\,000\) on 1 October 2025. With effect from November 2025, the bank charges him an interest of 2.5% for the remaining amount owed at the end of each month. Ivan makes a monthly repayment of \(\$x\) to the bank at the beginning of each month from 1 November 2025 onwards.
Show that the amount of money, in dollars, that Ivan owes after the repayment at the beginning of \(n\)th month is given by[3]
\(24\ 000(1.025^{n-1})-40x(1.025^n-1).\)
Suppose that the loan is repaid in 24 monthly repayments, find the value of the monthly repayment to the nearest cent and calculate the total interest paid on the loan.[3]
In which month and year will Ivan be able to repay the loan completely?[2]
It is now given that \(x=1500\).
Ivan found that if the bank were to charge him an interest of 7.5% for the remaining amount owed at the end of each month, the amount of money, in dollars, owed after the repayment at the beginning of \(n\)th month will be given by[1]
\(20\ 000+2500(1.075^{n-1}).\)
Determine if Ivan will be able to repay the loan completely.
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Answer:\(x=195\), \(y=70\); monthly repayment \(\(1309.18\), total interest \(\)7420.28\); 21 months, July 2027; the final proposed loan cannot be repaid