<p>Phil takes out a bank loan of $\$150\,000$ to buy a house, at an annual rate of $3.5\%$.</p>
<p>The interest is calculated at the end of each year and added to the amount outstanding.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the amount Phil would owe the bank after $20$ years. Give your answer to the nearest dollar.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>To pay off the loan, Phil makes an annual deposit of \$ $P$ at the end of every year in a savings account, paying an annual interest rate of $2$% . He makes his first deposit at the end of the first year after taking the loan.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Show that the total value of Phil's savings after $20$ years is \[\frac{(1.02^{20} - 1)P}{1.02 - 1}\]<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Given that Phil's aim is to own the house after $20$ years, find the value of $P$ to the nearest dollar.<span class="inline-marks">[2]</span></div></li></ol>
<p></p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">David visits a different bank and makes a single deposit \$ $Q$, the annual interest rate being $2.8$% .</div><ol class="i-list"><li><div class="list-item-paragraph">David wishes to withdraw \$ $5000$ at the end of each year for a period of $n$ years. Show that an expression for the minimum value of $Q$ is \[\frac{5000}{1.028} + \frac{5000}{1.028^2} + \cdots + \frac{5000}{1.028^n}\]</div></li><li><div class="list-item-paragraph">Hence, or otherwise, find the minimum value of $Q$ that would permit David to withdraw annual amounts of \$ $5000$ indefinitely. Give your answer to the nearest dollar.</div></li></ol></li></ol>

Phil takes out a bank loan of $\$150\,000$ to buy a house, at an annual rate of $3.5\%$.

The interest is calculated at the end of each year and added to the amount outstanding.

Find the amount Phil would owe the bank after $20$ years. Give your answer to the nearest dollar.

To pay off the loan, Phil makes an annual deposit of \$ $P$ at the end of every year in a savings account, paying an annual interest rate of $2$% . He makes his first deposit at the end of the first year after taking the loan.

Show that the total value of Phil's savings after $20$ years is \[\frac{(1.02^{20} - 1)P}{1.02 - 1}\]

Given that Phil's aim is to own the house after $20$ years, find the value of $P$ to the nearest dollar.

David visits a different bank and makes a single deposit \$ $Q$, the annual interest rate being $2.8$% .

Application of GP to financial problems

Application of GP to financial problems