Let's solve the sum $\sum_{j=2}^{8} (-3)^j$.

The expression $\sum_{j=2}^{8} (-3)^j$ means that we need to calculate the sum of the terms when $j$ takes values from 2 to 8 for the expression $(-3)^j$.

Step-by-Step Calculation

1. Calculate each term:

   (-3)^2 &= 9, \\ (-3)^3 &= -27, \\ (-3)^4 &= 81, \\ (-3)^5 &= -243, \\ (-3)^6 &= 729, \\ (-3)^7 &= -2187, \\ (-3)^8 &= 6561. \end{aligned}$$

2. Sum all the terms: $9 + (-27) + 81 + (-243) + 729 + (-2187) + 6561$

3. Perform the addition:

   9 - 27 &= -18, \\ -18 + 81 &= 63, \\ 63 - 243 &= -180, \\ -180 + 729 &= 549, \\ 549 - 2187 &= -1638, \\ -1638 + 6561 &= 4923. \end{aligned}$$

Final Answer

The value of the sum $\sum_{j=2}^{8} (-3)^j$ is $4923$.

Would you like further details or have any questions?

Here are some related questions to consider:

1. What is the general formula for the sum of a geometric series?
2. How would the sum change if the base of the exponent was different?
3. How can the properties of exponents help simplify the calculation of this sum?
4. How would the sum be affected if the upper limit of the summation was extended beyond 8?
5. What is the significance of alternating positive and negative terms in this sum?

Tip: When dealing with alternating series, it can be useful to pair positive and negative terms to simplify the calculation.