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.