The geometric mean of the series `1,2,4,8,16,....,2^n` is

Options

A. `2^(n+1//2)`
B. `2^(n+1)`
C. `2^(n//2)`
D. `2^(n)`

Answer

To find the geometric mean of the series \(1, 2, 4, 8, 16, \ldots, 2^n\), we can follow these steps:

### Step 1: Identify the series
The given series is \(1, 2, 4, 8, 16, \ldots, 2^n\). We can express this series in terms of powers of 2:
\[ 1 = 2^0, \quad 2 = 2^1, \quad 4 = 2^2, \quad 8 = 2^3, \quad 16 = 2^4, \ldots, \quad 2^n = 2^n \]
Thus, the series can be rewritten as:
\[ 2^0, 2^1, 2^2, 2^3, \ldots, 2^n \]

### Step 2: Use the formula for geometric mean
The geometric mean (GM) of a series of numbers \(a_1, a_2, a_3, \ldots, a_n\) is given by:
\[ GM = (a_1 \times a_2 \times a_3 \times \ldots \times a_n)^{\frac{1}{n}} \]
In our case, we have:
\[ GM = (2^0 \times 2^1 \times 2^2 \times \ldots \times 2^n)^{\frac{1}{n}} \]

### Step 3: Simplify the product
Since the bases are the same, we can combine the exponents:
\[ GM = 2^{(0 + 1 + 2 + \ldots + n) \cdot \frac{1}{n}} \]

### Step 4: Calculate the sum of exponents
The sum of the first \(n\) natural numbers is given by the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
Thus, we have:
\[ 0 + 1 + 2 + \ldots + n = \frac{n(n + 1)}{2} \]

### Step 5: Substitute the sum back into the GM formula
Now substituting the sum into the GM formula gives:
\[ GM = 2^{\left(\frac{n(n + 1)}{2}\right) \cdot \frac{1}{n}} = 2^{\frac{(n + 1)}{2}} \]

### Final Result
Thus, the geometric mean of the series \(1, 2, 4, 8, 16, \ldots, 2^n\) is:
\[ \boxed{2^{\frac{n + 1}{2}}} \]