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

The problem given is a summation:

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

This summation means we need to evaluate the expression \((-3)^j\) for each integer value of \(j\) from 2 to 8, and then add the results together.

Let's calculate this step by step:

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 \(j = 8\): \((-3)^8 = 6561\)

Now, sum all these values together:

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

Calculating the sum:

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

Thus, the value of the summation is **4923**.

Would you like more details or have any questions about this problem? 

Here are five related questions for further exploration:

1. What is the general form of a geometric series, and how does it apply to summations like this one?
2. How would the result change if the upper limit of the summation was increased to 9?
3. What is the significance of the alternating signs in the summation? How does this affect the overall sum?
4. Can you express this summation as a geometric series and use the geometric series formula to find the sum?
5. How would you approach this problem if the base was a fraction instead of an integer?

**Tip:** When dealing with summations, it’s useful to recognize if the terms form a geometric series, as this can simplify the process of finding the sum.

Evaluate the summation: \(\sum_{j=2}^{8} (-3)^j\)

This problem involves evaluating the summation \(\sum_{j=2}^{8} (-3)^j\), which covers key algebraic concepts such as summation and exponents. Step-by-step calculations lead to the final sum of 4923, suitable for students in Grades 9-12.

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