For a given positive integer [math]n[/math], a rate of interest [math]i[/math] can be found for which [math]4 a_{\overline{2 n} \mid i}=5 a_{\overline{n} \mid i}[/math].

Express in terms of [math]n[/math] how long it will take for money to double at this rate of interest.

• [math]\sqrt{n}[/math]
• [math]n/2[/math]
• [math]5n/8[/math]
• [math]n/\sqrt{2}[/math]
• [math]2n[/math]

References

Hlynka, Myron. "University of Windsor Old Tests 62-392 Theory of Interest". web2.uwindsor.ca. Retrieved November 23, 2023.

Solution: B

[math]4 \frac{1-v^{2 n}}{i}=5 \frac{1-v^n}{i}[/math] so [math]4 \frac{\left(1-v^n\right)\left(1+v^n\right)}{i}=5 \frac{1-v^n}{i}[/math]. If [math]v^n=1[/math] then [math]v=1[/math] so [math]i=1[/math] which cannot give [math]4 a_{\overline{2 n} \mid i}=5 a_{\bar{n} \mid i}[/math]. Thus [math]4\left(1+v^n\right)=5[/math] so [math]v^n=.25[/math] so [math](1+i)^n=4[/math] so [math](1+i)^{n / 2}=2[/math] at which point money is doubled. Thus [math]n / 2[/math] is the answer.

References

Hlynka, Myron. "University of Windsor Old Tests 62-392 Theory of Interest". web2.uwindsor.ca. Retrieved November 23, 2023.