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.