QUESTION $1(25$ MARKS$)$

$1\mathrm{.}1$ Show that $a_n^{}=\frac{1-v^n}{d}$

$1\mathrm{.}2$ A loan of $N$$$10,000$ is to be repaid over $10$ years by a level annuity payable

monthly in arrears. The amount of the monthly payment is calculated on the basis of

an interest rate of $1\%$ per month effective. Find the

$1\mathrm{.}2\mathrm{.}1$ Monthly repayment.

$1\mathrm{.}2\mathrm{.}2$ Total capital repaid and interest paid in the $1^{st}$ and last year respectively.

$1\mathrm{.}2\mathrm{.}3$ After which monthly repayment the outstanding loan is first

less than $N$$$5,000.$

$1\mathrm{.}2\mathrm{.}4$ For which monthly repayment the capital repaid first exceeds the interest.