The arthmetic mean of the numbers
`1,3,3^(2), … ,3^(n-1),` is

Options

A. `(3^(n)-1)/(2)`
B. `(3^(n)-1)/(2n)`
C. `(3^(n)+1)/(2)`
D. `(3^(n)+1)/(2n)`

Explanation

To find the arithmetic mean of the numbers \(1, 3, 3^2, \ldots, 3^{n-1}\), we can follow these steps:

### Step 1: Identify the series
The given numbers are \(1, 3, 3^2, \ldots, 3^{n-1}\). This is a geometric progression (GP) where:
- The first term \(A = 1\)
- The common ratio \(R = 3\)

### Step 2: Determine the number of terms
The number of terms in this series is \(n\) (from \(3^0\) to \(3^{n-1}\)).

### Step 3: Calculate the sum of the GP
The formula for the sum \(S_n\) of the first \(n\) terms of a geometric progression is given by:
\[ S_n = A \frac{R^n - 1}{R - 1} \]
Substituting the values:
- \(A = 1\)
- \(R = 3\)
- \(n = n\)

We get:
\[ S_n = 1 \cdot \frac{3^n - 1}{3 - 1} = \frac{3^n - 1}{2} \]

### Step 4: Calculate the arithmetic mean
The arithmetic mean \(M\) is given by the formula:
\[ M = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{S_n}{n} \]
Substituting the sum we calculated:
\[ M = \frac{\frac{3^n - 1}{2}}{n} = \frac{3^n - 1}{2n} \]

### Final Result
Thus, the arithmetic mean of the numbers \(1, 3, 3^2, \ldots, 3^{n-1}\) is:
\[ \frac{3^n - 1}{2n} \]

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