The geometric mean of numbers `7, 7^(2),7^(3),…,7^(n),` is

Options

A. `7^(7//4)`
B. `7^(4//7)`
C. `7^((n-1)/(2))`
D. `7^((n+1)/(2))`

Answer

To find the geometric mean of the numbers \( 7, 7^2, 7^3, \ldots, 7^n \), we can follow these steps:

### Step 1: Identify the Numbers
The numbers given are:
- \( a_1 = 7 \)
- \( a_2 = 7^2 \)
- \( a_3 = 7^3 \)
- ...
- \( a_n = 7^n \)

### Step 2: Write the Formula for Geometric Mean
The formula for the geometric mean \( GM \) of \( n \) numbers \( a_1, a_2, \ldots, a_n \) is given by:
\[ GM = (a_1 \cdot a_2 \cdot a_3 \cdots a_n)^{\frac{1}{n}} \]

### Step 3: Substitute the Values
Substituting the values of \( a_1, a_2, \ldots, a_n \):
\[ GM = (7 \cdot 7^2 \cdot 7^3 \cdots \cdot 7^n)^{\frac{1}{n}} \]

### Step 4: Combine the Exponents
Since the bases are the same, we can combine the exponents:
\[ GM = (7^{1 + 2 + 3 + \ldots + n})^{\frac{1}{n}} \]

### Step 5: Calculate the Sum of the Exponents
The sum of the first \( n \) natural numbers is given by the formula:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
Thus, we can substitute this into our expression:
\[ GM = (7^{\frac{n(n + 1)}{2}})^{\frac{1}{n}} \]

### Step 6: Simplify the Expression
Now, simplifying the expression:
\[ GM = 7^{\frac{n(n + 1)}{2n}} = 7^{\frac{n + 1}{2}} \]

### Final Result
Therefore, the geometric mean of the numbers \( 7, 7^2, 7^3, \ldots, 7^n \) is:
\[ \boxed{7^{\frac{n + 1}{2}}} \]

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