Amanita, let's first confirm the problem:

Problem Statement:
You are given an arithmetic sequence (등차수열) where the 3rd term is 9 and the 10th term is 23. You are asked to find the 20th term of this sequence.

Let's break down the steps to solve this problem:

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Step-by-Step Solution Plan

1. Recall the formula for the nth term of an arithmetic sequence:
   The nth term, $a_n$, is given by:

$a_n = a_1 + (n-1)d$
where $a_1$ is the first term and $d$ is the common difference.

2. Set up equations using the given terms:

   • The 3rd term: $a_3 = a_1 + 2d = 9$
   • The 10th term: $a_{10} = a_1 + 9d = 23$

3. Solve the system of equations to find $a_1$ and $d$.

4. Use the values of $a_1$ and $d$ to find the 20th term:

$a_{20} = a_1 + 19d$

5. Check which answer choice matches the result.

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Now, let's execute each step in detail.

Step 1 & 2: Set up the equations

We have:

• $a_1 + 2d = 9$
• $a_1 + 9d = 23$

Step 3: Solve for $a_1$ and $d$

Let's solve this system of equations. We have solved the system and found:

• The first term $a_1 = 5$
• The common difference $d = 2$

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Step 4: Find the 20th term

Now, let's use the formula for the nth term:

$$a_{20} = a_1 + 19d$$

Substitute the values:

$$a_{20} = 5 + 19 \times 2$$

Let's calculate the value. The 20th term of the arithmetic sequence is:

$$a_{20} = 43$$

So, the answer is $\boxed{43}$, which matches choice ②.

Summary of steps:

1. Used the formula for the nth term of an arithmetic sequence.
2. Set up and solved two equations to find the first term and common difference.
3. Substituted these values into the formula for the 20th term.
4. Calculated the answer as 43.

If you have any more questions about arithmetic sequences, feel free to ask, Amanita!