A Financial Advisor $($FA$)$ told your sister if she invests in the stock market, $$100$ dollars a month during the next $40$ years, she will have $$1$ Million for her retirement. The FA said to her that a rate of $10\%$ is a feasible expecting rate because the S&P Index shows an average of $12\%$ during the past $40$ years. Because you are studying Finance, your sister asked for your help figuring if that is true. $(10$ points$)$