https://file.aiquickdraw.com/mathbot/file/1725384827642yeyhbcid.jpg

The expression provided is a summation, specifically:

\[
\sum_{j=2}^{8} (-3)^j
\]

This means we need to calculate the sum of \((-3)^j\) for \(j\) ranging from 2 to 8. Let's compute this step by step.

### Step-by-Step Calculation:

1. **For \( j = 2 \):**
   \[
   (-3)^2 = 9
   \]

2. **For \( j = 3 \):**
   \[
   (-3)^3 = -27
   \]

3. **For \( j = 4 \):**
   \[
   (-3)^4 = 81
   \]

4. **For \( j = 5 \):**
   \[
   (-3)^5 = -243
   \]

5. **For \( j = 6 \):**
   \[
   (-3)^6 = 729
   \]

6. **For \( j = 7 \):**
   \[
   (-3)^7 = -2187
   \]

7. **For \( j = 8 \):**
   \[
   (-3)^8 = 6561
   \]

### Summing All Terms:
Now, we add up all these values:

\[
9 + (-27) + 81 + (-243) + 729 + (-2187) + 6561
\]

### Calculating the Sum:
Let's compute the total:

\[
9 - 27 + 81 - 243 + 729 - 2187 + 6561 = 4923
\]

So, the value of the sum is **4923**.

Would you like more details on this process, or do you have any questions?

---

**Relative Questions:**

1. What is the significance of alternating signs in the summation of powers of a negative number?
2. How does the exponent affect the behavior of the base \(-3\) in the sequence?
3. Can this summation be represented as a geometric series? If so, how?
4. What would happen if the lower limit of summation changed from \( j = 2 \) to \( j = 1 \)?
5. How do different bases affect the overall sum when calculated similarly?

**Tip:** Always check whether the series or sum has a pattern (like geometric progression), as it can simplify the calculations.

Evaluate the summation from j=2 to j=8 of (-3) raised to the power j.

Learn how to evaluate the summation of (-3) raised to the power j for j ranging from 2 to 8 with a detailed step-by-step solution. This problem covers algebraic concepts such as exponents and summation. Suitable for students in Grades 9-11.

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