We are tasked with evaluating the summation:

$\sum_{j=1}^{5} (5j - 6)$

This means we need to calculate the expression $5j - 6$ for each value of $j$ from 1 to 5, and then sum the results.

Step 1: Evaluate each term

• For $j = 1$: $5(1) - 6 = 5 - 6 = -1$
• For $j = 2$: $5(2) - 6 = 10 - 6 = 4$
• For $j = 3$: $5(3) - 6 = 15 - 6 = 9$
• For $j = 4$: $5(4) - 6 = 20 - 6 = 14$
• For $j = 5$: $5(5) - 6 = 25 - 6 = 19$

Step 2: Sum the results

Now sum all the evaluated terms:

$-1 + 4 + 9 + 14 + 19 = 45$

Thus, the value of the summation is:

$\boxed{45}$

Would you like more details or explanations on any of the steps?

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Here are five related questions to extend your understanding:

1. How do you calculate a summation with a general formula?
2. What happens if the upper limit of the summation increases to 10?
3. Can you generalize the formula for summing $(5j - 6)$ for any upper limit $n$?
4. How would you compute the sum if the expression inside was quadratic, like $5j^2 - 6$?
5. How do arithmetic sequences relate to summations like this?

Tip: Always check each term carefully when performing summations to avoid small arithmetic mistakes!