Find the accumulated amount after 7 years if $14,000 is invested at 3%/year compounded in the following ways. (Round your answers to the nearest cent.)
(a) compounded annually
$ 


(b) compounded semiannually
$ 


(c) compounded quarterly
$ 


(d) compounded monthly
$ 


To find the accumulated amount, we can use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $A$ is the accumulated amount (future value),
- $P$ is the principal (initial investment),
- $r$ is the annual interest rate (as a decimal),
- $n$ is the number of times the interest is compounded per year,
- $t$ is the number of years.

We are given:
- $P = 14,000$,
- $r = 0.03$ (since 3% interest),
- $t = 7$ years.

### (a) Compounded annually:
For annual compounding, $n = 1$.

\[
A = 14000 \left(1 + \frac{0.03}{1}\right)^{1 \times 7} = 14000 (1.03)^7
\]

Let me calculate that.

### (b) Compounded semiannually:
For semiannual compounding, $n = 2$.

\[
A = 14000 \left(1 + \frac{0.03}{2}\right)^{2 \times 7} = 14000 \left(1 + 0.015\right)^{14}
\]

I will calculate this next.

### (c) Compounded quarterly:
For quarterly compounding, $n = 4$.

\[
A = 14000 \left(1 + \frac{0.03}{4}\right)^{4 \times 7} = 14000 \left(1 + 0.0075\right)^{28}
\]

I will calculate that too.

### (d) Compounded monthly:
For monthly compounding, $n = 12$.

\[
A = 14000 \left(1 + \frac{0.03}{12}\right)^{12 \times 7} = 14000 \left(1 + 0.0025\right)^{84}
\]

Now, let me calculate each of these values.The accumulated amounts after 7 years for each compounding method are:

- (a) Compounded annually: **\$17,218.23**
- (b) Compounded semiannually: **\$17,244.58**
- (c) Compounded quarterly: **\$17,257.96**
- (d) Compounded monthly: **\$17,266.97**

Would you like more details or have any further questions? Here are 5 related questions to deepen your understanding:

1. How does the frequency of compounding affect the final amount?
2. What would be the accumulated amount if the interest rate was 5% instead of 3%?
3. How does continuous compounding differ from the methods mentioned above?
4. What is the effect of doubling the time to 14 years with the same interest rate?
5. How much interest is earned in each case after 7 years?

**Tip:** The more frequently interest is compounded, the higher the accumulated amount due to the interest being applied on previously earned interest more often.

Find the accumulated amount after 7 years if $14,000 is invested at 3%/year compounded in the following ways. (Round your answers to the nearest cent.)
(a) compounded annually
$ 
(b) compounded semiannually
$ 
(c) compounded quarterly
$ 
(d) compounded monthly
$

Learn how to calculate the accumulated amount when $14,000 is invested at a 3% annual interest rate compounded annually, semiannually, quarterly, and monthly. This step-by-step guide includes key concepts such as compound interest, exponential growth, and interest rate calculations. Suitable for high school students in Grades 10-12.

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