Find the interest rate for a ​$8500 deposit accumulating to ​$12 comma 077​, compounded annually for 7 years.
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Part 1
The interest rate is 
  
enter your response here​%.
​(Do not round until the final answer. Then round to two decimal places as​ needed.)

To solve for the interest rate, we'll use the formula for compound interest:

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

Where:
- $ A $ is the amount of money accumulated after $ n $ years, including interest.
- $ P $ is the principal amount (the initial money).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested or borrowed for, in years.

Given values:
- $ A = 12,077 $
- $ P = 8,500 $
- $ n = 1 $ (compounded annually)
- $ t = 7 $ years

We need to solve for $ r $. The formula simplifies to:

\[
12,077 = 8,500 \times \left(1 + r\right)^{7}
\]

First, divide both sides by 8,500:

\[
\frac{12,077}{8,500} = \left(1 + r\right)^{7}
\]

\[
1.4206 = \left(1 + r\right)^{7}
\]

Next, take the 7th root of both sides:

\[
(1 + r) = 1.4206^{\frac{1}{7}}
\]

\[
1 + r \approx 1.05003
\]

Finally, subtract 1 from both sides to find $ r $:

\[
r \approx 1.05003 - 1
\]

\[
r \approx 0.05003
\]

Convert $ r $ to a percentage:

\[
r \approx 5.00\%
\]

So, the interest rate is **5.00%**.

Would you like more details or have any questions? 

Here are some related questions you might be interested in:
1. How does changing the number of compounding periods per year affect the interest rate?
2. What is the difference between simple interest and compound interest?
3. How would the interest rate change if the time period was different?
4. What happens to the accumulated amount if the interest rate increases?
5. Can the compound interest formula be used for continuous compounding?

**Tip:** When solving for the interest rate using compound interest, taking the appropriate root is key to isolating $ r $.

Learn how to calculate the interest rate for a $8500 deposit compounded annually over 7 years, resulting in an accumulated amount of $12,077. Step-by-step solution using the compound interest formula. Suitable for high school students.

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