`"Evaluate "Sigma_(k=1)^(11) (2+3^(k))`

Explanation

To evaluate the expression \( \Sigma_{k=1}^{11} (2 + 3^k) \), we can break it down into two parts: the sum of the constant term and the sum of the geometric series.

### Step 1: Separate the summation
We can rewrite the summation as:
\[ \Sigma_{k=1}^{11} (2 + 3^k) = \Sigma_{k=1}^{11} 2 + \Sigma_{k=1}^{11} 3^k \]

### Step 2: Calculate the first summation
The first part, \( \Sigma_{k=1}^{11} 2 \), is simply adding the number 2 for each value of \( k \) from 1 to 11. Since there are 11 terms:
\[ \Sigma_{k=1}^{11} 2 = 2 \times 11 = 22 \]

### Step 3: Calculate the second summation
The second part, \( \Sigma_{k=1}^{11} 3^k \), is a geometric series where the first term \( a = 3^1 = 3 \) and the common ratio \( r = 3 \). The number of terms \( n = 11 \).

The sum of a geometric series can be calculated using the formula:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values:
\[ S_{11} = 3 \frac{3^{11} - 1}{3 - 1} = 3 \frac{3^{11} - 1}{2} \]

### Step 4: Combine the results
Now we can combine both parts:
\[ \Sigma_{k=1}^{11} (2 + 3^k) = 22 + 3 \frac{3^{11} - 1}{2} \]

### Step 5: Simplify the expression
To simplify further:
\[ = 22 + \frac{3(3^{11} - 1)}{2} \]
\[ = 22 + \frac{3^{12} - 3}{2} \]

### Final Answer
Thus, the final result is:
\[ \Sigma_{k=1}^{11} (2 + 3^k) = 22 + \frac{3^{12} - 3}{2} \]