The average (arithmetic mean) of
`3^(30), 3^(60)` and `3^(90)` is

Options

A. `3^(27) + 3^(57) + 3^(87)`
B. `3^29x37`
C. `3^(29) + 3^(59) + 3^(89)`
D. `3^(177)`

Answer

To find the average of the numbers \(3^{30}\), \(3^{60}\), and \(3^{90}\), we can follow these steps:

### Step 1: Write down the formula for the average
The average (arithmetic mean) of a set of numbers is given by the formula:
\[ \text{Average} = \frac{\text{Sum of the numbers}}{\text{Number of values}} \]
In this case, we have three numbers.

### Step 2: Calculate the sum of the numbers
We need to calculate the sum of \(3^{30}\), \(3^{60}\), and \(3^{90}\):
\[ \text{Sum} = 3^{30} + 3^{60} + 3^{90} \]

### Step 3: Factor out the common term
Notice that \(3^{30}\) is a common factor in all three terms. We can factor it out:
\[ \text{Sum} = 3^{30} (1 + 3^{30} + 3^{60}) \]

### Step 4: Substitute the sum into the average formula
Now we substitute the sum back into the average formula:
\[ \text{Average} = \frac{3^{30} (1 + 3^{30} + 3^{60})}{3} \]

### Step 5: Simplify the expression
We can simplify the average:
\[ \text{Average} = 3^{30 - 1} (1 + 3^{30} + 3^{60}) = 3^{29} (1 + 3^{30} + 3^{60}) \]

### Step 6: Further simplify the expression inside the parentheses
The expression inside the parentheses can be simplified further:
\[ 1 + 3^{30} + 3^{60} = 1 + 3^{30} + (3^{30})^2 = 1 + x + x^2 \quad \text{(where \(x = 3^{30}\))} \]
This is a quadratic expression in terms of \(x\).

### Step 7: Final expression for the average
Thus, the average can be expressed as:
\[ \text{Average} = 3^{29} (1 + 3^{30} + 3^{60}) \]

### Conclusion
The average of \(3^{30}\), \(3^{60}\), and \(3^{90}\) is:
\[ \text{Average} = 3^{29} (1 + 3^{30} + 3^{60}) \]