Sam sets up a savings account with an initial deposit of $1000$ in Year $1.$

Each year, the bank pays $2\%$ interest on the previous year's balance into the

savings account. As an illustration, the account balance in Year $1,2,$ and $3$

are shown in the table below:

$($a$)$ Let us denote the account balance in Year $n$ to be $x_n.$ Find the values

of the first term $x_1=$a and the common ratio $r,$ and write down the

recurrence system for $(x_n).$

$($b$)$ Find the closed form for $x_n.$

$($c$)$ Calculate Sam's account balance in Year $10,$ up to two decimal places.

$($d$)$ In what year will the account balance first exceed $1500?$