You provided a sequence defined by the following recurrence relation:

• $a_1 = 3$
• $a_n = a_{n-1} + 11$

This is an arithmetic sequence because each term after the first is generated by adding a constant difference (11 in this case) to the previous term.

General Form of the Sequence

For an arithmetic sequence, the $n$th term can be expressed as:

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

Where:

• $a_1$ is the first term.
• $d$ is the common difference.
• $n$ is the term number.

Applying the Values

Here, $a_1 = 3$ and $d = 11$, so:

$a_n = 3 + (n-1) \times 11$

Simplifying the equation:

$a_n = 3 + 11n - 11$ $a_n = 11n - 8$

So, the general form of the sequence is:

$\boxed{a_n = 11n - 8}$

Example Calculation

To find specific terms:

• $a_2 = 11(2) - 8 = 22 - 8 = 14$
• $a_3 = 11(3) - 8 = 33 - 8 = 25$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you find the sum of the first $n$ terms in an arithmetic sequence?
2. What are the properties of an arithmetic sequence?
3. How would you solve for the term number $n$ given $a_n$?
4. Can you derive the formula for the general term of a geometric sequence?
5. How do arithmetic sequences differ from geometric sequences?

Tip: Always double-check the common difference to ensure it's consistent throughout the sequence when dealing with arithmetic sequences.