Confusingly Continuously Compounding Interest Rates

Samantha is in dire need of money and asked her friend Sarah to lend her some money, $1000 to be exact. After finally persuading her, Sarah finally agrees but gives her a really weird interest rate.

Sarah: "You know what a compound interest rate is?"

Samantha: "Sure. It's basically including interest rate when calculating interest rate. Say you're going to lend me with a monthly compound interest. Then, every month, the total value added to my debt would be the product of my total debt (money initially loaned plus all interests accumulated) and the interest rate. Formula is $P_{\text{new}} = P_{\text{old}} \left(1+\frac{r}{n}\right)^t$ if I recall correctly."

Sarah: "Correct. Then do you know what continuously compounded interest rate is?"

Samantha: "Well, it's like compound interest rate, but instead of compounding every month, you compound every single instant of time, right? So take $n$ to $\infty$ , which using some high-school calculus would be just $P_{\text{new}} = P_{\text{old}} e^{rt}$ ."

Sarah: "Good. So I want to change you continuously compounding interest rate, but with the interest rate changing. I'll lend you for exactly two years. At any point of time $t$ , your annual interest rate will be $\sqrt{t}$ . So initially interest rate is just $0\%$ . At 6 months, it'll be $\sqrt{0.5} = 70.7\%$ . At the end of the first year, it'll be a full $100\%$ . At the end of the second, $\sqrt{2}=141\%$ "

Samantha was initially confused, but she agreed anyways, figuring that it couldn't be too bad, right?

What will her total debt be at the end of 2 years?

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In the formulas Samantha gives,

• $P_{\text{new}}$ is the debt at the end of the loan period, in dollars
• $P_{\text{old}}$ is the debt at the start of the loan period, in dollars
• $r$ is the annual interest rate
• $n$ is the number of times an interest rate is compounded
• $t$ is the total length of the loan period, in years
• $e$ is Euler's constant