https://file.aiquickdraw.com/mathbot/file/1727071580850esds2mmy.jpg

Let's break down the two problems presented in the image.

### Problem 1
The problem asks to calculate the following summation:

\[
\sum_{x=1}^{12} x(x + 1)
\]

We need to evaluate this summation by expanding the expression \( x(x + 1) \) for each \( x \) from 1 to 12.

### Problem 2
This one involves a different type of summation. Given the following values:
\[
x_1 = 7, \, x_2 = 13, \, x_3 = 14, \, x_4 = 17, \, x_5 = 19, \, x_6 = 22
\]

We are asked to calculate:

\[
\sum_{i=1}^{4} (x_i - 2)
\]

For this, we will sum the values of \( x_i - 2 \) for the first four values of \( x_i \) provided.

Let me solve these two problems now.### Solution to Problem 1
The value of the summation:

\[
\sum_{x=1}^{12} x(x + 1) = 728
\]

### Solution to Problem 2
The value of the summation:

\[
\sum_{i=1}^{4} (x_i - 2) = 43
\]

Would you like more details on these solutions, or have any questions about the process?

Here are 5 related questions you can explore:
1. How would the result of Problem 1 change if the upper limit were extended to \( x = 15 \)?
2. Can the expression \( x(x + 1) \) be simplified using algebraic identities before summing?
3. How does changing the constant value \( 2 \) in Problem 2 affect the summation result?
4. What is the general formula for the sum of \( x(x + 1) \) from 1 to \( n \)?
5. How would you compute the sum of the squares of the values \( x_i \)?

**Tip:** Summation formulas like \( \sum_{x=1}^{n} x \) or \( \sum_{x=1}^{n} x^2 \) can be powerful tools for quickly calculating large sums.

1. Calcula el valor numérico de la siguiente sumatoria: \sum_{x=1}^{12} x(x+1). 2. Considerando que x_1=7, x_2=13, x_3=14, x_4=17, x_5=19, y x_6=22, calcule: \sum_{i=1}^{4} (x_i - 2).

Learn how to solve summations involving arithmetic series and sequences. This problem covers calculating \sum_{x=1}^{12} x(x+1) and summing sequence values with adjustments such as \sum_{i=1}^{4} (x_i - 2). Suitable for high school students in Grades 10-12.

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