(i) Proof in worked solution
(ii) 24 years
(i) Quarterly rate \(=\dfrac{6}{100} \div 4=0.015\)
\begin{aligned} A_{1} &=1000000(1.015)^{4}-800000 \\ A_{2} &=A_{1}(1.015)^{4}-80000 \\ &=\left[1000000(1.015)^{4}-80000](1.015)^{4}-80000\right.\\ &=1000000(1.015)^{8}-80000(1.015)^{4}-80000 \\ &=1000000(1.015)^{8}-80000\left(1+1.015^{4}\right) \end{aligned}
(ii)
\begin{aligned} A_{n} &=1000000(1.015)^{4 n}-80000\left(1+1.015^{4} + 1.015^{8}+\ldots+1.015^{4(n-1)}\right) \\ &=1000000(1.015)^{4 n}-80000 \times \frac{\left(1.015^{4 n}-1\right)}{1.015^{4}-1} \end{aligned}
When \(A_n=0\)
\begin{aligned} 1000000(1.015)^{4 n} &=80000 \frac{\left(1.015^{4 n}-1\right)}{1.015^{4}-1} \\ &=1303705.5\left(1.015^{4 n}-1\right) \end{aligned}
Let \(1.015^{4n} = y\)
\begin{aligned}1000000 y&=1303705.5 y-1303705.5 \\303705.5 y&=1303705.5 \\y&=4.292663 \\1.015^{4 n}&=4.292663 \\4 n \log 1.015&=\log 4.292663 \\n&=24.46 \\n=24 \text { years }\end{aligned}
