To solve this problem, we will use the formula for the nth term of an arithmetic progression (AP):

$T_n = a + (n - 1) \cdot d$

Where:

• $T_n$ is the nth term,
• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Given Information:

• 3rd term ($T_3$) = 18
• 9th term ($T_9$) = 30

We need to find the sum of the 20th term. To do this, we need to first determine the values of $a$ (the first term) and $d$ (the common difference).

Step 1: Write two equations for the 3rd and 9th terms:

$T_3 = a + (3 - 1) \cdot d = 18 \quad \Rightarrow \quad a + 2d = 18 \quad \text{(Equation 1)}$ $T_9 = a + (9 - 1) \cdot d = 30 \quad \Rightarrow \quad a + 8d = 30 \quad \text{(Equation 2)}$

Step 2: Solve the system of equations:

Subtract Equation 1 from Equation 2:

$(a + 8d) - (a + 2d) = 30 - 18$ $6d = 12$ $d = 2$

Step 3: Substitute $d = 2$ into Equation 1:

$a + 2 \cdot 2 = 18$ $a + 4 = 18$ $a = 14$

Step 4: Find the 20th term:

Now that we have $a = 14$ and $d = 2$, we can find the 20th term using the formula for the nth term:

$T_{20} = a + (20 - 1) \cdot d$ $T_{20} = 14 + 19 \cdot 2$ $T_{20} = 14 + 38 = 52$

Final Answer:

The 20th term of the arithmetic progression is 52.

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Would you like further clarification or details on this solution? Here are some related questions to explore:

1. How do you derive the general formula for the nth term of an AP?
2. What happens if the common difference is negative in an AP?
3. How would you find the sum of the first $n$ terms of an AP?
4. How does the sum of terms in an AP change if the common difference is doubled?
5. What is the significance of the first term and common difference in the behavior of an AP?

Tip: When solving AP-related problems, always make sure to define the first term $a$ and the common difference $d$ early on, as they are key to calculating any term in the sequence.