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.