Question

you deposit $2000 in an account that pays 6% interest compounded semiannually. after 4 years, the interest rate is increased to 6.08% compounded quarterly. what will be the value of the account after a total of 8 years? click the icon to view some finance formulas. the value of the account will be $ (round to the nearest dollar as needed.)

Explanation:

Step1: Calculate value after first 4 years (semiannual compounding)

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $n$ is the number of times interest is compounded per year, and $t$ is the number of years.

For the first 4 years:

• $P=\$2000$
• $r = 6\%=0.06$
• $n = 2$ (compounded semiannually)
• $t = 4$

Substitute these values into the formula:
$A_1=2000(1 +\frac{0.06}{2})^{2\times4}=2000(1 + 0.03)^{8}$

Calculate $(1 + 0.03)^{8}$:
$(1.03)^{8}\approx1.266770081$

Then $A_1 = 2000\times1.266770081\approx\$2533.54$

Step2: Calculate value after next 4 years (quarterly compounding)

Now, the principal for the next 4 - year period is $P_2=A_1\approx\$2533.54$

• $r = 6.08\% = 0.0608$
• $n = 4$ (compounded quarterly)
• $t = 4$

Use the compound - interest formula again: $A_2=P_2(1+\frac{r}{n})^{nt}$

Substitute the values:
$A_2 = 2533.54(1+\frac{0.0608}{4})^{4\times4}=2533.54(1 + 0.0152)^{16}$

Calculate $(1 + 0.0152)^{16}$:
$(1.0152)^{16}\approx1.27510444$

Then $A_2=2533.54\times1.27510444\approx\$3230$ (rounded to the nearest dollar)

Answer:

$\$3230$