$Q1.[20$ Marks$]$

$(a)[4$ Marks$]$ Using $(Ia{)}_{\frac{?}{\text{b}\text{a}\text{r}}(n)|}=\frac{a_{\frac{?}{\text{b}\text{a}\text{r}}(n)|}^{}-n{v}^n}{i},$ show that

$(Da{)}_{\frac{?}{\text{b}\text{a}\text{r}}(n)|}=\frac{n-a_{\frac{?}{\text{b}\text{a}\text{r}}(n)}}{i}$

$(b)[16$ Marks$]$ A loan $istobe$ repaid over $10$ years $by$ means $of$ half$-$yearly instalments payable $in$ arrears.

The amount $of$ the first instalment $is1,500$ and each subsequent instalment increases $by100$ for

the first $5$ years with the instalment $at$ the end $of$ year $5$ being $2,400.$ After that each instalment

decreases $by50.$

The rate $of$ interest charged $by$ the lender $is6\%$ per annum convertible half$-$yearly.

$(i)[7$ Marks$]$ Calculate the initial loan amount.

$(ii)[7$ Marks$]$ Determine the interest and capital components $of$ the $7th$ instalment.

$(iii)[2$ Marks$]$ Suppose that the initial amount $of$ the loan $isasin$ part $(a)$ but the half$-$yearly

repayments are $of2016$ each throughout the term. Determine, with justification, whether

the total interest paid under this revised schedule would $be$ greater $or$ less than that paid under

the original repayment schedule. $no$ excel $no$ chatgpt step $by$ step guide $on$ paper please