Find the general term of an arithmetic sequence whose 9th term is 26

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:
  \[
  a_9 = a_1 + 8d = 26
  \]
  
This equation contains two unknowns (\(a_1\) and \(d\)). To find the general term, we need either an additional piece of information, such as another term or the common difference. However, with only the 9th term given, the general term can be expressed in terms of the unknowns \(a_1\) and \(d\).

### Expressing the General Term

We already know that:

\[
a_9 = a_1 + 8d = 26
\]

So, the first term can be written as:

\[
a_1 = 26 - 8d
\]

Substituting this back into the general formula \(a_n = a_1 + (n-1)d\):

\[
a_n = (26 - 8d) + (n-1)d
\]

Simplifying:

\[
a_n = 26 - 8d + nd - d
\]

\[
a_n = nd - 9d + 26
\]

\[
a_n = d(n - 9) + 26
\]

### Conclusion

The general term of the arithmetic sequence is:

\[
\boxed{a_n = d(n - 9) + 26}
\]

This formula expresses the general term \(a_n\) in terms of the common difference \(d\).

Would you like further details or have any questions?

Here are five related questions:
1. What happens to the sequence if the common difference \(d\) is negative?
2. How does knowing a second term help in fully determining the general term?
3. How do you find the sum of the first \(n\) terms of this sequence if \(d\) is given?
4. Can you find the common difference if the first term \(a_1\) is known?
5. How does the sequence change as \(d\) increases or decreases?

**Tip:** When given limited information about a sequence, expressing the general term in terms of unknowns can still provide valuable insight into the structure of the sequence.

Learn how to find the general term of an arithmetic sequence when given the 9th term as 26. Understand the formula for the nth term of an arithmetic sequence and its application. Suitable for high school students studying arithmetic sequences.

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