Watch the video and then solve the problem given below.
Click here to watch the video.
Use the compound interest formulas and to solve the problem given. Round answers to the nearest cent.
Find the accumulated value of an investment of for 4 years at an interest rate of if the money is a. compounded semiannually, b. compounded quarterly, c.
compounded month; compounded continuously.
a. What is the accumulated value if the money is compounded semiannually?
(Round your answer to the nearest cent.)

Ask by Barnett Christensen. in the United States

Mar 23,2025

Question

Watch the video and then solve the problem given below.
Click here to watch the video.
Use the compound interest formulas and to solve the problem given. Round answers to the nearest cent.
Find the accumulated value of an investment of for 4 years at an interest rate of if the money is a. compounded semiannually, b. compounded quarterly, c.
compounded month; compounded continuously.
a. What is the accumulated value if the money is compounded semiannually?
(Round your answer to the nearest cent.)

Ask by Barnett Christensen. in the United States

Mar 23,2025

UpStudy AI · Accepted Answer

Let

$$
\mathrm{P}=25000,\quad \mathrm{r}=0.055,\quad t=4,\quad \text{and for semiannual compounding } n=2.
$$

Using the compound interest formula

$$
\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{r}}{n}\right)^{nt},
$$

substitute the values:

$$
\mathrm{A}=25000\left(1+\frac{0.055}{2}\right)^{2\cdot 4} = 25000\left(1+0.0275\right)^8 = 25000\,(1.0275)^8.
$$

We now compute $$(1.0275)^8$$. Step by step multiplication (approximate values):

1. $$1.0275^1 \approx 1.0275$$,
2. $$1.0275^2\approx 1.0275\times1.0275=1.05675625$$,
3. $$1.0275^3\approx 1.05675625\times1.0275\approx 1.086989$$,
4. $$1.0275^4\approx 1.086989\times1.0275\approx 1.117227$$,
5. $$1.0275^5\approx 1.117227\times1.0275\approx 1.147485$$,
6. $$1.0275^6\approx 1.147485\times1.0275\approx 1.178779$$,
7. $$1.0275^7\approx 1.178779\times1.0275\approx 1.210124$$,
8. $$1.0275^8\approx 1.210124\times1.0275\approx 1.242539.$$

Multiplying by the principal:

$$
\mathrm{A}\approx25000\times1.242539\approx31063.49.
$$

Rounding to the nearest cent, the accumulated value is

$$
\boxed{\$31,\!063.49}.
$$
The accumulated value of the investment compounded semiannually is \$31,063.49.

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