Let's find the geometric mean (g) of the given numbers: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.

The geometric mean of a set of n positive numbers is the n-th root of their product.

The given sequence is a geometric progression where the first term is 2 and the common ratio is 2. The terms are $2^1, 2^2, 2^3, \dots, 2^{10}$.

There are 10 numbers in the sequence (from $2^1$ to $2^{10}$). So, n = 10.

The geometric mean (g) is given by the formula:

$$g = \sqrt[n]{a_1 \times a_2 \times \dots \times a_n}$$

In this case, n = 10 and the numbers are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.

$$g = \sqrt[10]{2 \times 4 \times 8 \times 16 \times 32 \times 64 \times 128 \times 256 \times 512 \times 1024}$$

We can write each number as a power of 2:

$$g = \sqrt[10]{2^1 \times 2^2 \times 2^3 \times 2^4 \times 2^5 \times 2^6 \times 2^7 \times 2^8 \times 2^9 \times 2^{10}}$$

Using the property of exponents ($a^m \times a^n = a^{m+n}$), we can add the powers of 2:

The sum of the exponents is $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10$.

This is the sum of the first 10 natural numbers, which can be calculated using the formula $\frac{n(n+1)}{2}$ where n=10.

Sum of exponents $= \frac{10(10+1)}{2} = \frac{10 \times 11}{2} = \frac{110}{2} = 55$.

So, the product of the numbers is $2^{55}$.

Now, substitute this back into the geometric mean formula:

$$g = \sqrt[10]{2^{55}}$$

Using the property of roots and exponents ($\sqrt[n]{a^m} = a^{m/n}$):

$$g = 2^{55/10} = 2^{5.5}$$

We need to determine which range this value falls into. We can rewrite $2^{5.5}$ as $2^5 \times 2^{0.5}$.

$$g = 2^5 \times \sqrt{2}$$

We know that $2^5 = 32$.

So, $g = 32 \times \sqrt{2}$.

We know that the value of $\sqrt{2}$ is approximately 1.414.

Let's estimate the value of g:

$$g \approx 32 \times 1.414$$

Calculating the product:

• $32 \times 1 = 32$
• $32 \times 0.4 = 12.8$
• $32 \times 0.01 = 0.32$
• $32 \times 0.004 = 0.128$

Adding these values: $32 + 12.8 + 0.32 + 0.128 = 45.248$.

So, $g \approx 45.248$.

Now let's compare this value with the given ranges:

1. $8 < g < 16$: $8 < 45.248 < 16$. This is false.
2. $16 < g < 32$: $16 < 45.248 < 32$. This is false.
3. $32 < g < 64$: $32 < 45.248 < 64$. This is true.
4. $g > 64$: $45.248 > 64$. This is false.

Alternatively, consider the range bounds: $32 = 2^5$ and $64 = 2^6$. The geometric mean is $g = 2^{5.5}$. Since $5 < 5.5 < 6$, it follows that $2^5 < 2^{5.5} < 2^6$, which means $32 < g < 64$. This confirms our estimate.

Therefore, the correct range for the geometric mean g is $32 < g < 64$.

Geometric Mean Calculation Summary

Understanding the Geometric Mean

The geometric mean is a type of average that is often used for sets of positive numbers that are interpreted in terms of their products or compounded values, or for data that follows a geometric progression.

• It is calculated by multiplying all the numbers in the set and then taking the nth root of the product, where n is the count of numbers in the set.
• It is particularly useful for calculating average rates of growth or change.

Comparing Geometric Mean with Other Means

For a set of positive numbers (not all equal), the harmonic mean, geometric mean, and arithmetic mean have the following relationship:

Harmonic Mean $\le$ Geometric Mean $\le$ Arithmetic Mean

Let's quickly calculate the arithmetic mean (AM) for the given sequence to compare.

The sequence is 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.

Sum $= 2 + 4 + \dots + 1024$. This is a geometric series with $a=2$, $r=2$, $n=10$.

Sum $= a \frac{r^n - 1}{r - 1} = 2 \frac{2^{10} - 1}{2 - 1} = 2 \frac{1024 - 1}{1} = 2 \times 1023 = 2046$.

AM $= \frac{\text{Sum}}{\text{n}} = \frac{2046}{10} = 204.6$.

We found the geometric mean $g \approx 45.248$. As expected, $g < AM$ (45.248 < 204.6).

Conclusion on the Geometric Mean Range

Based on the calculation $g = 2^{5.5} \approx 45.248$, we confirmed that g falls within the range specified by $32 < g < 64$. This is because $32 = 2^5$ and $64 = 2^6$, and $5 < 5.5 < 6$.

Revision Table: Key Concepts

Additional Information: Geometric Mean Properties

• The geometric mean is always less than or equal to the arithmetic mean (Equality holds only if all numbers are identical).
• It is sensitive to small values; a single zero value makes the geometric mean zero, regardless of the other values. This is why it's typically defined for positive numbers.
• It is used in finance (e.g., calculating average investment returns over multiple periods), biology (e.g., averaging growth rates), and other fields.