To find the twentieth term of the arithmetic sequence, we need to determine the common difference $d$ and the first term $a_1$ of the sequence. The general formula for the $n$-th term of an arithmetic sequence is:

$a_n = a_1 + (n - 1)d$

Step 1: Use the information given

• The $14$-th term ($a_{14}$) is 30: $a_{14} = a_1 + (14 - 1)d = a_1 + 13d = 30$

• The $21$-st term ($a_{21}$) is 44: $a_{21} = a_1 + (21 - 1)d = a_1 + 20d = 44$

Step 2: Solve the system of equations

From the two equations:

1. $a_1 + 13d = 30$
2. $a_1 + 20d = 44$

Subtract the first equation from the second to eliminate $a_1$: $(a_1 + 20d) - (a_1 + 13d) = 44 - 30$ $7d = 14 \quad \implies \quad d = 2$

Substitute $d = 2$ into the first equation: $a_1 + 13(2) = 30$ $a_1 + 26 = 30 \quad \implies \quad a_1 = 4$

Step 3: Find the 20th term

The formula for the $n$-th term is: $a_n = a_1 + (n - 1)d$

For the $20$-th term ($n = 20$): $a_{20} = a_1 + (20 - 1)d = 4 + 19(2)$ $a_{20} = 4 + 38 = 42$

Final Answer:

The twentieth term of the arithmetic sequence is: $\boxed{42}$

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Related Questions:

1. What is the sum of the first 20 terms of this arithmetic sequence?
2. How can you derive the general term ($a_n$) for this sequence?
3. What would the sequence look like from the first term to the twenty-first term?
4. How would the calculation change if $a_{14}$ and $a_{21}$ were reversed?
5. What happens to the sequence if the common difference ($d$) is changed to 3?

Tip:

In arithmetic sequences, knowing any two terms and their positions allows you to calculate everything about the sequence, including the common difference and any other term.