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The question asks you to calculate the average annual gain over the last five years, specifically the geometric mean of the percentage increases. Here's how we can calculate it:

The stock price increases for the five years are:  
2%, 6%, 8%, 10%, 13%.

We calculate the geometric mean using the formula:

\[
\text{Geometric Mean} = \left( \prod_{i=1}^{n} (1 + r_i) \right)^{\frac{1}{n}} - 1
\]
Where:
- \( r_i \) are the individual percentage increases, converted to decimals.
- \( n \) is the number of years (which is 5).

Let's convert the percentages to decimals and calculate the geometric mean:
- Year 1: \( 1 + \frac{2}{100} = 1.02 \)
- Year 2: \( 1 + \frac{6}{100} = 1.06 \)
- Year 3: \( 1 + \frac{8}{100} = 1.08 \)
- Year 4: \( 1 + \frac{10}{100} = 1.10 \)
- Year 5: \( 1 + \frac{13}{100} = 1.13 \)

Now, multiply these factors:

\[
1.02 \times 1.06 \times 1.08 \times 1.10 \times 1.13 = 1.437227
\]

Next, take the 5th root of this product:

\[
(1.437227)^{\frac{1}{5}} = 1.07531
\]

Finally, subtract 1 and convert back to a percentage:

\[
1.07531 - 1 = 0.07531 \quad \text{or} \quad 7.53\%
\]

So, the average annual gain (geometric mean) is **7.53%**.

Would you like further clarification or details on this?  

Here are 5 related questions to explore:
1. How is the geometric mean different from the arithmetic mean?
2. Why is the geometric mean more appropriate for stock price growth?
3. How would the result change if there were a negative return in one of the years?
4. What are the limitations of using the geometric mean in financial contexts?
5. How can compounding affect investment returns over time?

**Tip:** When comparing returns over time, always consider the impact of compounding, especially for volatile assets.

Suppose a certain company stock has its share price rise by the following percentages during each of the last 5 years: 2%, 6%, 8%, 10%, 13%. Find the average annual gain over the last five years (i.e., the geometric mean). Round your answer to two decimal places.

Learn how to calculate the average annual gain for a company's stock price over five years using the geometric mean. This problem involves percentage growth rates and exponential functions, suitable for high school students.

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