https://file.aiquickdraw.com/mathbot/file/1730214283265rlrytmoz.jpg

Let's go through each question in the image and solve it step-by-step.

### Question 3
You invest $2600 in an account with an Annual Percentage Rate (APR) of 9%. We are asked to calculate the investment value after 3 years with different compounding frequencies.

The general formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $ A $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount (initial investment).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times interest is compounded per year.
- $ t $ is the number of years.

Given values:
- $ P = 2600 $
- $ r = 0.09 $ (since 9% = 0.09)
- $ t = 3 $

#### (a) Compounded Yearly
For yearly compounding, $ n = 1 $.

\[
A = 2600 \left(1 + \frac{0.09}{1}\right)^{1 \times 3} = 2600 \times (1.09)^3
\]
Calculating:
\[
A \approx 2600 \times 1.295029 = 3367.08
\]

So, the answer should be **$3367.08** (rounded to the nearest cent).

#### (b) Compounded Monthly
For monthly compounding, $ n = 12 $.

\[
A = 2600 \left(1 + \frac{0.09}{12}\right)^{12 \times 3} = 2600 \times \left(1 + 0.0075\right)^{36}
\]
Calculating:
\[
A \approx 2600 \times 1.349353 = 3508.32
\]

So, the answer should be **$3508.32** (rounded to the nearest cent).

### Question 4
Assume an investment of $6000 earns an APR of 6% compounded monthly, and we are asked for the account balance and interest after 14 months.

Given values:
- $ P = 6000 $
- $ r = 0.06 $
- $ t = \frac{14}{12} = 1.1667 $ years (since 14 months is 1.1667 years)
- $ n = 12 $ (monthly compounding)

#### Account Balance After 14 Months
\[
A = 6000 \left(1 + \frac{0.06}{12}\right)^{12 \times 1.1667} = 6000 \times (1 + 0.005)^{14}
\]
Calculating:
\[
A \approx 6000 \times 1.072058 = 6432.35
\]

So, the account balance after 14 months should be **$6432.35** (rounded to the nearest cent).

#### Interest Earned After 14 Months
To find the interest earned, subtract the initial principal from the total amount:
\[
\text{Interest} = A - P = 6432.35 - 6000 = 432.35
\]

So, the interest earned after 14 months is **$432.35** (rounded to the nearest cent).

---

If you'd like, I can explain any of the steps in more detail or answer additional questions related to this material.

### Relative Questions
1. How does increasing the compounding frequency affect the final amount?
2. How would the result change if the APR was different?
3. What is the formula for continuously compounded interest?
4. Why does monthly compounding yield a higher balance than yearly compounding?
5. What would the balance be after 5 years with the same conditions?

### Tip
Remember that more frequent compounding periods lead to slightly higher accumulated amounts due to interest-on-interest effects.

You invest $2600 in an account that pays an APR of 9%. The problem asks for the value of the investment after 3 years with different compounding frequencies: compounded yearly and compounded monthly. Additionally, there's a question about an investment of $6000 with an APR of 6%, compounded monthly, asking for the account balance and interest after 14 months.

Learn how to calculate compound interest for investments with varying compounding frequencies. This example involves an investment of $2600 with a 9% APR compounded yearly and monthly over 3 years, as well as an investment of $6000 with a 6% APR compounded monthly over 14 months. Suitable for high school students.

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