$($a$)$ For a given positive integer $n,$ a rate of interest i can be found such that $4s_{2n}=9s_n.$

Express in terms of $n,$ how long it will take for money to double at this rate of

interest.

$($b$)$ An annuity is payable half$-$yearly for six years, the first half$-$yearly payment of

amount RM$1800$ being due after two years. The amount of subsequent payments

decreases by RM$30$ each half$-$year. On the basis of nominal interest of $5\%$

convertible half$-$yearly, calculate the present value of the annuity.

$($c$)$ An annuity$-$immediate is payable annually for $20$ years. The annual payment is

RM$50$ for six years, followed by RM$70$ for nine years and finally RM$100$ for five

years.

$($i$)$ Show that the value of the annuity at the time of the first payment can be

written as:

$100a_{20}^{}-30a_{15}^{}-20a_6^{}$

$($ii$)$ Show that the value of the annuity at the time of the last payment can be

written as:

$50s_{\frac{?}{\text{b}\text{a}\text{r}}(20)}+20s_{14}+30s_{\frac{?}{\text{b}\text{a}\text{r}}(5)}$