Question $(9)$

Let $g_n$ denote the size of a tumor $($in mg $)$ on the nth day of an experiment following the start of chemotherapy. Assume that the experiment starts on day $n=1.$ If theses daily sizes form a geometric sequence $\{g_n\},$ where

$g_n=g_1{r}^{n-1}$

is the general nth term and

$($i$)$ The sum of the tumor sizes on the $4^{th}$ day and $8^{th}$ day equal $2$ mg $,$ and

$($ii$)$ The sum of the tumor sizes on the $5^{th}$ and $9^{th}$ day equal $1$ mg $.$

$($a$)$ Find the proportion $r$ by which the size of the tumor decreased each day.

$($b$)$ Find the size of the tumor on the first day of the start of chemotherapy.

Note: Do not calculate the final answer.