For a given positive integer [math]n[/math], a rate of interest [math]i[/math] can be found such that [math]4s_{\overline{2n}|} = 9s_{\overline{n}|}[/math]. Express in terms of [math]n[/math] how long it will take for money to double at this rate of interest.

• 3n
• 3.05n
• 3.11n
• 3.15n
• 3.22n

References

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

Solution: C

We need [math]t[/math] such that [math](1+i)^t=2[/math] so [math]t=\ln 2 /(\ln (1+i))[/math]. But [math]4 s_{\overline{2 n} \mid}=9 s_{\bar{n} \mid}[/math] so [math]4 \frac{(1+i)^{2 n}-1}{i}=[/math] [math]9 \frac{(1+i)^n-1}{i}[/math] so [math]4 \frac{\left((1+i)^n-1\right)\left((1+i)^n+1\right)}{i}=9 \frac{(1+i)^n-1}{i}[/math] so [math](1+i)^n+1=9 / 4[/math] so [math](1+i)^n=5 / 4=1.25[/math]. Thus [math]n \ln (1+i)=\ln 1.25[/math]. Finally [math]t=\ln 2 /(\ln (1+i))=n \ln 2 / \ln 1.25=3.106 n[/math].

References

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