Answer:
Elizabeth earns $ 154.86 due to interests from the beginning to the fourth year and $ 62.75 due to interest only in the fourth year.
Step-by-step explanation:
Elizabeth deposits each year the same amount, which becomes greater due to composite interest. From statement, we notice the following formulas of recurrence:
Initial amount ([tex]t = 0[/tex])
[tex]C = C_{o}[/tex] (1)
First year ([tex]t = 1[/tex])
[tex]C_{1} = C_{o}\cdot (1+r)+C_{o}[/tex] (2)
Second year ([tex]t = 2[/tex])
[tex]C_{2} = C_{1}\cdot (1+r)+C_{o}[/tex] (3)
Third year ([tex]t = 3[/tex])
[tex]C_{3} = C_{2}\cdot (1+r)+C_{o}[/tex] (4)
Fourth year ([tex]t = 4[/tex])
[tex]C_{4} = C_{2}\cdot (1+r)+C_{o}[/tex] (5)
Where:
[tex]C_{o}[/tex] - Initial amount of Elizabeth's account, measured in US dollars.
[tex]r[/tex] - Interest rate ratio, dimensionless.
If we know that [tex]r = 0.02[/tex] and [tex]C_{o} = \$\,750[/tex], then the interest earned by Elizabeth in the 4th year is:
First year
[tex]C_{r,1} = r\cdot C_{o}[/tex] (6)
[tex]C_{r,1} = (0.02)\cdot (\$\,750)[/tex]
[tex]C_{r,1} = \$\,15[/tex]
Second year
[tex]C_{1} = C_{o}\cdot (1+r)+C_{o}[/tex]
[tex]C_{r,2} = r\cdot C_{1}[/tex] (7)
[tex]C_{1} = (\$\,750)\cdot (1+0.02)+\$\,750[/tex]
[tex]C_{1} = \$ 1515[/tex]
[tex]C_{r,2} = (0.02)\cdot (\$\,1515)[/tex]
[tex]C_{r,2} = \$\,30.3[/tex]
Third year
[tex]C_{2} = C_{1}\cdot (1+r)+C_{o}[/tex]
[tex]C_{r,3} = r\cdot C_{2}[/tex] (8)
[tex]C_{2} = (\$\,1515)\cdot (1+0.02)+\$\,750[/tex]
[tex]C_{2} = \$\,2340.75[/tex]
[tex]C_{r,3} = (0.02)\cdot (\$\,2340.75)[/tex]
[tex]C_{r,3} = \$\,46.81[/tex]
Fourth year
[tex]C_{3} = C_{2}\cdot (1+r) +C_{o}[/tex]
[tex]C_{r,4} = r\cdot C_{3}[/tex] (9)
[tex]C_{3} = (\$\,2340.75)\cdot (1+0.02)+\$\,750[/tex]
[tex]C_{3} = \$\,3137.56[/tex]
[tex]C_{r,4} = (0.02)\cdot (\$\,3137.56)[/tex]
[tex]C_{r,4} = \$\,62.75[/tex]
The money gained due to interest is determined by the following sum, that is:
[tex]C_{r} = C_{r,1}+C_{r,2}+C_{r,3}+C_{r,4}[/tex]
[tex]C_{r} = \$\,15+\$\,30.30+\$\,46.81+\$\,62.75[/tex]
[tex]C_{r} = \$\,154.86[/tex]
Elizabeth earns $ 154.86 due to interests from the beginning to the fourth year and $ 62.75 due to interest only in the fourth year.
