The geometric mean of the given growth factors $1.085, 1.041, 1.026$ is calculated as:

$\text{Geometric Mean} = \sqrt[3]{1.085 \times 1.041 \times 1.026}$

Let's compute this value. Given growth factors growth_factors = [1.085, 1.041, 1.026]

Compute geometric mean

geometric_mean = (growth_factors[0] * growth_factors[1] * growth_factors[2]) ** (1/3)

Round to four decimal places

geometric_mean_rounded = round(geometric_mean, 4) geometric_mean_roundedThe average growth factor, or geometric mean, is 1.0504 (rounded to four decimal places).

Part 2: Equivalent Single Percentage Growth Rate

The equivalent single percentage growth rate is found by converting the geometric mean back to a percentage:

$\text{Growth Rate} = (\text{Geometric Mean} - 1) \times 100$

Let's compute this value. Compute the equivalent single percentage growth rate equivalent_growth_rate = (geometric_mean_rounded - 1) * 100 equivalent_growth_rateThe equivalent single percentage growth rate is 5.04%.

Part 3: Comparing with the Arithmetic Mean

Now, let's compare this result with the arithmetic mean of the given percentages:

$\text{Arithmetic Mean} = \frac{8.5 + 4.1 + 2.6}{3}$

Let's compute this value. Given growth percentages growth_percentages = [8.5, 4.1, 2.6]

Compute arithmetic mean

arithmetic_mean = sum(growth_percentages) / len(growth_percentages) arithmetic_meanThe arithmetic mean of the given percentages is 5.07% (rounded to two decimal places).

Conclusion

The geometric mean growth rate (5.04%) is slightly lower than the arithmetic mean (5.07%), which is expected because the geometric mean accounts for compounding effects and provides a more accurate measure for growth rates over time.

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Would you like a more detailed explanation of any step?

Here are some related questions:

1. How is the geometric mean different from the arithmetic mean in financial analysis?
2. When should the geometric mean be used instead of the arithmetic mean?
3. Can the geometric mean ever be greater than the arithmetic mean?
4. How does the geometric mean help in calculating average investment returns?
5. What happens if one of the values in the geometric mean calculation is zero?

Tip: The geometric mean is particularly useful in cases involving percentages, ratios, or rates of change, as it preserves the proportionality between values.