A small business takes out a loan of $$10,000$ for initial inventory. The interest rate on the loan is $5\%$ compounded annually.

$($a$)$ Write down the first five terms of the geometric progression representing the total amount the business owes after each year, including the initial loan amount.

$($b$)$ What is the explicit formula for the nth term $($total amount owed after $n$ years$)$ of this GP$?$

$($c$)$ The business owner wants to pay back the loan within $3$ years. How much will the total amount owed be after $3$ years?

$($d$)$ Imagine the business is successful and decides to reinvest its profits each year, earning a constant annual return of $8\%$ interest compounded annually. If they start with an initial investment of $$5,000,$ what will their total investment be after $4$ years using the formula for the sum of a geometric series?

$($e$)$ Financial advisors often recommend compound interest as a powerful tool for wealth creation. Briefly explain, in the context of the reinvestment scenario $($part d$),$ why the concept of geometric progression is significant for long$-$term financial planning.