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The image shows a multiple-choice math question that asks for the value of \(\sum_{i=1}^{4} 2^{i} \) and provides the following options:

- A) 240
- B) 30
- C) 780
- D) 255

### Solution:
The given sum is \(\sum_{i=1}^{4} 2^{i} \), which means we need to calculate the sum of the first four powers of 2.

\[
\sum_{i=1}^{4} 2^{i} = 2^1 + 2^2 + 2^3 + 2^4
\]

Let's calculate each term:
- \(2^1 = 2\)
- \(2^2 = 4\)
- \(2^3 = 8\)
- \(2^4 = 16\)

Now, add these values together:

\[
2 + 4 + 8 + 16 = 30
\]

### Answer:
The correct answer is **B) 30**.

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. How do you generalize the sum of a series like \(\sum_{i=1}^{n} 2^{i} \)?
2. What is the formula for the sum of a geometric series?
3. How does the sum \(\sum_{i=1}^{4} 2^{i}\) compare to \(\sum_{i=0}^{4} 2^{i}\)?
4. How can you calculate powers of 2 quickly without a calculator?
5. How would the sum change if we used a different base, say 3 instead of 2?
6. What are some common applications of geometric series in real-world problems?
7. How can you express the sum \(\sum_{i=1}^{n} 2^{i}\) using a closed-form formula?
8. What is the significance of exponential growth in different mathematical contexts?

**Tip:** When dealing with powers of numbers in a series, always look out for patterns that can simplify calculations.

Learn how to calculate the sum of powers of 2, specifically \( \sum_{i=1}^{4} 2^{i} \), with a detailed step-by-step solution. This problem covers arithmetic series and exponential growth concepts. Suitable for students in Grades 9-12.

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