Find the general term of an arithmetic sequence whose 9th term is 26 and 20th term is 59.

To find the general term of an arithmetic sequence, we use the formula for the \(n\)th term:

\[
a_n = a_1 + (n-1)d
\]

where:
- \(a_n\) is the \(n\)th term,
- \(a_1\) is the first term,
- \(d\) is the common difference, and
- \(n\) is the term number.

Given:
- The 9th term (\(a_9\)) is 26, so:
  \[
  a_9 = a_1 + 8d = 26 \quad \text{(Equation 1)}
  \]
- The 20th term (\(a_{20}\)) is 59, so:
  \[
  a_{20} = a_1 + 19d = 59 \quad \text{(Equation 2)}
  \]

### Step 1: Subtract Equation 1 from Equation 2

\[
(a_1 + 19d) - (a_1 + 8d) = 59 - 26
\]

\[
11d = 33
\]

\[
d = 3
\]

### Step 2: Substitute \(d = 3\) back into Equation 1

\[
a_1 + 8(3) = 26
\]

\[
a_1 + 24 = 26
\]

\[
a_1 = 2
\]

### Step 3: Write the General Term

Now that we know \(a_1 = 2\) and \(d = 3\), we can write the general term:

\[
a_n = 2 + (n-1) \times 3
\]

Simplifying:

\[
a_n = 2 + 3n - 3 = 3n - 1
\]

So, the general term of the arithmetic sequence is:

\[
\boxed{a_n = 3n - 1}
\]

Would you like further details or have any questions?

Here are five related questions:
1. How do you find the common difference if only two terms of an arithmetic sequence are given?
2. What is the first term of the sequence, and how does it influence the general term?
3. How would the formula change if the 9th term and 20th term were negative?
4. How can you find the sum of the first \(n\) terms of this sequence?
5. What is the significance of the general term in identifying sequence patterns?

**Tip:** Always double-check by plugging the general term back into the conditions provided to ensure accuracy.

Learn how to find the general term of an arithmetic sequence when given specific terms. This problem involves calculating the general term using the nth term formula of an arithmetic sequence. Suitable for students in Grades 9-12.

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