To find and add up an arithmetic sequence in a problem, here are some simple steps to follow:

Steps to Identify an Arithmetic Sequence:

1. What is an Arithmetic Sequence?
   An arithmetic sequence is a list of numbers where the difference between one number and the next is always the same. This difference is called the common difference ($d$).

2. Find the First Term:
   Look for the first number in the sequence, which we call $a_1$. For example, in the sequence 2, 5, 8, ..., the first number is $2$.

3. Figure Out the Common Difference:
   To find the common difference, subtract the first number from the second number. In our example, $d = 5 - 2 = 3$.

4. Use the General Formula:
   You can find any term in the sequence using this formula:
   $a_n = a_1 + (n - 1) d$
   Here, $n$ is the term number you want to find.

Steps to Find the Sum:

5. Sum Formula:
   To add up the first $n$ terms of an arithmetic sequence, you can use this formula:
   $S_n = \frac{n}{2} (a_1 + a_n)$
   or
   $S_n = \frac{n}{2} (2a_1 + (n - 1)d)$

6. Identify $n$:
   Figure out how many terms $n$ you want to add together.

7. Calculate the Last Term:
   If you need to, find the last term $a_n$ using the general formula.

8. Compute the Sum:
   Plug the numbers you found into the sum formula to get $S_n$.

Example:

Let’s look at the arithmetic sequence 3, 7, 11,... and add up the first 15 terms ($n = 15$):

• The first term, $a_1 = 3$

• The common difference, $d = 4$

• To find the 15th term:
  $a_{15} = 3 + (15 - 1) \cdot 4 = 3 + 56 = 59$

• Now, to find the sum:
  $S_{15} = \frac{15}{2} (3 + 59)$
  $S_{15} = \frac{15}{2} \cdot 62 = 15 \cdot 31 = 465$

So, the total sum of the first 15 terms is 465!