https://file.aiquickdraw.com/mathbot/file/1730364635086ai80pdjy.jpg

The expression given is:

\[
\sum_{i=2}^{7} 3^i
\]

To compute this, we evaluate \( 3^i \) for each integer \( i \) from 2 to 7, and then sum those results.

1. When \( i = 2 \): \( 3^2 = 9 \)
2. When \( i = 3 \): \( 3^3 = 27 \)
3. When \( i = 4 \): \( 3^4 = 81 \)
4. When \( i = 5 \): \( 3^5 = 243 \)
5. When \( i = 6 \): \( 3^6 = 729 \)
6. When \( i = 7 \): \( 3^7 = 2187 \)

Now, summing these values:

\[
9 + 27 + 81 + 243 + 729 + 2187 = 3276
\]

Thus, the result of the summation is:

\[
\sum_{i=2}^{7} 3^i = 3276
\]

Would you like more details on the calculation process or have any questions?

Here are some related questions to explore further:
1. How would you compute a summation with a different base, such as \( 5^i \) instead of \( 3^i \)?
2. What is the general formula for the sum of a geometric sequence?
3. How would the result change if the range was \( i=1 \) to \( i=7 \) instead?
4. How would you calculate the sum if the exponent was variable, like \( 3^{i+1} \) instead of \( 3^i \)?
5. What would happen if you used a different lower limit, such as \( i=3 \)?

**Tip:** In summation problems with exponential terms, identify if it forms a geometric sequence to simplify calculations.

Explore the summation of 3^i for i ranging from 2 to 7. This problem involves evaluating exponential growth and applying the concept of a geometric series. Ideal for students in Grades 10-12, with detailed steps showing each term's calculation and the final sum of 3276.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation