Let's break down and solve each of these mathematical scenarios step by step.

Given values:

$x_1 = 5, \quad x_2 = 15, \quad x_3 = 3, \quad x_4 = 3, \quad x_5 = 8, \quad x_6 = 10, \quad x_7 = 12$

Now, let's calculate the value of each expression.

1. Expression 1: $\sum_{i=1}^{n} x_i$

This represents the summation of the values from $x_1$ to $x_n$. Assuming $n = 7$, we will sum all values of $x_i$ from $x_1$ to $x_7$:

$\sum_{i=1}^{7} x_i = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7$

$= 5 + 15 + 3 + 3 + 8 + 10 + 12 = 56$

So, the value of $\sum_{i=1}^{7} x_i = 56$.

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2. Expression 2: $\sum_{j=2}^{7} x_j + 6$

This represents the summation of the values from $x_2$ to $x_7$, and then adding 6 to the result:

$\sum_{j=2}^{7} x_j + 6 = x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + 6$

$= 15 + 3 + 3 + 8 + 10 + 12 + 6 = 57$

So, the value of $\sum_{j=2}^{7} x_j + 6 = 57$.

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3. Expression 3: $x_5$

This is simply the value of $x_5$, which is 8.

So, $x_5 = 8$.

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4. Expression 4: $\sum_{i=3}^{6} x_i$

This represents the summation of the values from $x_3$ to $x_6$:

$\sum_{i=3}^{6} x_i = x_3 + x_4 + x_5 + x_6$

$= 3 + 3 + 8 + 10 = 24$

So, the value of $\sum_{i=3}^{6} x_i = 24$.

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Summary of Results:

1. $\sum_{i=1}^{7} x_i = 56$
2. $\sum_{j=2}^{7} x_j + 6 = 57$
3. $x_5 = 8$
4. $\sum_{i=3}^{6} x_i = 24$

Do you need further details or have any questions about the process?

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Here are 5 related questions for further exploration:

1. What would $\sum_{i=1}^{5} x_i$ be, using the same values?
2. How does changing the index limits in a summation affect the final result?
3. What happens if you increase the value of $x_5$ in the above expressions?
4. Can you calculate $\sum_{i=2}^{4} x_i + 10$?
5. What is the difference between the sum of $x_1$ through $x_7$ and the sum of $x_3$ through $x_6$?

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Tip: Always check the limits of your summations carefully. Different index ranges lead to different results even when using the same set of numbers!