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We are given an arithmetic sequence where the general term is defined as:

\[
u_n = 3n - 7
\]

We need to find the sum of the terms from \( u_7 \) to \( u_{19} \), i.e.,

\[
\sum_{i=7}^{19} u_i
\]

### Step 1: Calculate the terms of the sequence

We know that the general term \( u_n = 3n - 7 \). So, we can compute the values of \( u_7 \) and \( u_{19} \):

- \( u_7 = 3(7) - 7 = 21 - 7 = 14 \)
- \( u_{19} = 3(19) - 7 = 57 - 7 = 50 \)

### Step 2: Find the number of terms

The number of terms from \( u_7 \) to \( u_{19} \) is:

\[
n = 19 - 7 + 1 = 13
\]

### Step 3: Apply the formula for the sum of an arithmetic sequence

The sum \( S \) of an arithmetic sequence is given by the formula:

\[
S = \frac{n}{2} \times (u_{\text{first}} + u_{\text{last}})
\]

Where:
- \( n = 13 \) (the number of terms),
- \( u_{\text{first}} = u_7 = 14 \),
- \( u_{\text{last}} = u_{19} = 50 \).

Substituting these values into the formula:

\[
S = \frac{13}{2} \times (14 + 50) = \frac{13}{2} \times 64 = 13 \times 32 = 416
\]

### Final Answer:

The sum of the terms from \( u_7 \) to \( u_{19} \) is:

\[
\sum_{i=7}^{19} u_i = 416
\]

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Would you like further details or explanations? Here are some related questions:

1. How do you find the sum of any arithmetic series?
2. What is the significance of the common difference in arithmetic sequences?
3. How does the formula for the sum of an arithmetic series change for different starting terms?
4. Can the general term formula \( u_n = 3n - 7 \) be used to find negative terms if \( n \) is less than 1?
5. What happens if we want the sum of a geometric sequence instead?

**Tip:** When dealing with arithmetic sequences, always remember the simple pattern in differences between terms!

An arithmetic sequence is given by u_n = 3n - 7. Find the sum from i = 7 to i = 19.

Solve the arithmetic sequence u_n = 3n - 7 by finding the sum from i = 7 to i = 19. This solution covers key arithmetic concepts, including finding the general term and using the formula for the sum of a series. Suitable for students in Grades 9-11.

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