Thank you for the clarification. Let's correct the solution based on \( n = 10 \) instead of \( n = 11 \).

### Problem Recap with Corrected Information
You're given an arithmetic progression (AP) where:
- The first term \( a_1 = 5 \)
- The common difference \( d = 3 \)
- The nth term of the progression is denoted by \( u_n \)

You need to find the sum of the sequence from \( n = 10 \) to \( n = 20 \), i.e.,

\[
\sum_{n=10}^{20} u_n
\]

### Step 1: General Formula for the nth Term
As derived earlier, the nth term \( u_n \) is given by:

\[
u_n = 3n + 2
\]

### Step 2: Sum of the Terms from \( n = 10 \) to \( n = 20 \)
The sum of terms from \( u_{10} \) to \( u_{20} \) can be found using the sum formula for an arithmetic progression:

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

where:
- \( k \) is the number of terms
- The first term is \( u_{10} \)
- The last term is \( u_{20} \)

#### Number of Terms:
\( k = 20 - 10 + 1 = 11 \)

#### First Term \( u_{10} \):
\[
u_{10} = 3(10) + 2 = 30 + 2 = 32
\]

#### Last Term \( u_{20} \):
\[
u_{20} = 3(20) + 2 = 60 + 2 = 62
\]

#### Sum \( S \):
\[
S = \frac{11}{2} \times (32 + 62)
\]
\[
S = \frac{11}{2} \times 94 = 11 \times 47 = 517
\]

### Final Answer
The sum \( \sum_{n=10}^{20} u_n \) is **517**.

Would you like any further details or have any questions?

### Relative Questions:
1. How do you determine the first term of a specific segment in an arithmetic sequence?
2. How does the number of terms \( k \) affect the sum of an arithmetic series?
3. How would this problem change if the common difference \( d \) were a fraction?
4. What if the sequence started at a negative term—how would that affect the sum?
5. How can you generalize the sum of terms between any two indices \( m \) and \( n \)?

### Tip:
Always double-check the starting and ending indices in summation problems, as even small mistakes in interpreting these can lead to incorrect results.

Discover how to calculate the sum of an arithmetic progression from n = 10 to n = 20 using the formula for the nth term and the sum formula of an arithmetic series. This problem involves finding the sum of terms with a given first term, common difference, and specific range of indices. Suitable for students in Grades 10-12.

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