Sum of first n natural numbers is given by `(n(n+1))/(2)`. What is the geometric mean of the series 1,2,4,8 . . . `2^(n)` ?

Options

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

Answer

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

### Step 1: Identify the series
The given series can be expressed as:
\[ 1, 2, 4, 8, \ldots, 2^n \]
This series can also be written in the form of powers of 2:
\[ 2^0, 2^1, 2^2, 2^3, \ldots, 2^n \]

### Step 2: Count the number of terms
The series has \(n + 1\) terms (from \(2^0\) to \(2^n\)).

### Step 3: Use the formula for the geometric mean
The geometric mean \(G\) of a set of numbers is given by the formula:
\[ G = (a_1 \times a_2 \times \ldots \times a_k)^{1/k} \]
where \(a_1, a_2, \ldots, a_k\) are the terms in the series and \(k\) is the number of terms.

### Step 4: Calculate the product of the terms
The product of the terms in the series is:
\[ 2^0 \times 2^1 \times 2^2 \times \ldots \times 2^n \]
This can be simplified using the properties of exponents:
\[ = 2^{0 + 1 + 2 + \ldots + n} \]

### Step 5: Calculate the sum of the exponents
The sum of the first \(n\) natural numbers is given by the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
Thus, the sum of the exponents \(0 + 1 + 2 + \ldots + n\) is:
\[ \frac{n(n + 1)}{2} \]

### Step 6: Substitute back into the product
Therefore, the product of the terms becomes:
\[ 2^{\frac{n(n + 1)}{2}} \]

### Step 7: Calculate the geometric mean
Now, substituting back into the formula for the geometric mean:
\[ G = \left(2^{\frac{n(n + 1)}{2}}\right)^{\frac{1}{n + 1}} \]
This simplifies to:
\[ G = 2^{\frac{n(n + 1)}{2(n + 1)}} \]
\[ = 2^{\frac{n}{2}} \]

### Step 8: Final result
Thus, the geometric mean of the series \(1, 2, 4, 8, \ldots, 2^n\) is:
\[ G = 2^{\frac{n}{2}} \]