Kathryn deposits 100 into an account at the beginning of each 4-year period for 40 years. The account credits interest at an annual effective interest rate of i. The accumulated amount in the account at the end of 40 years is X, which is 5 times the accumulated amount in the account at the end of 20 years.

Calculate X.

• 6,195
• 6,300
• 6,385
• 6,415
• 6,487

References

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

Solution: A

There are 10 payments at times [math]0,4,8, \ldots, 36[/math]. Let [math]j[/math] be the interest rate for a 4 year period and let [math]i[/math] be the annual rate. Then [math](1+j)=(1+i)^4[/math]. Thus [math]100 \ddot{s}_{\overline{10} \mid j}=5(100) \ddot{s}_{\overline{5} \mid j}[/math]. So

[[math]]100 \frac{(1+j)^{10}-1}{j}(1+j)=5(100) \frac{(1+j)^5-1}{j}(1+j)[[/math]]

. Simplifying, using [math](1+j)^{10}-1=\left((1+j)^5-1\right)\left((1+j)^5+1\right)[/math] gives [math]\left((1+j)^5+1\right)=5[/math] so [math](1+j)^5=4[/math] so

[[math]]X=100 \frac{(1+j)^{10}-1}{j}(1+j)=100 \frac{4^2-1}{4^{1 / 5}-1} 4^{1 / 5}=6194.72[[/math]]

References

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