To find patterns in sequences and series through problem-solving, students can use some helpful strategies:

1. Recognizing Types of Sequences: Start by figuring out if the sequence is arithmetic, geometric, or something else.

   • An arithmetic sequence has a steady difference between each number. For example, in the sequence 2, 4, 6, 8, the difference is 2.
   • A geometric sequence has a consistent ratio. For example, in the sequence 2, 4, 8, 16, the ratio is 2 because each number is multiplied by 2.

2. Analyzing Terms: Look at the first few numbers in the sequence to see if you can find a pattern. Finding the differences or ratios between the numbers can be really helpful.

   • For example, in the sequence 2, 4, 8, 16, if you divide one number by the previous one, you see it’s always 2, showing it’s geometric.

3. General Formulas: Once you spot a pattern, create a formula for the $n$-th term. You can use formulas like:

   • For an arithmetic series, the sum can be calculated as $S_n = \frac{n}{2}(a_1 + a_n)$.
   • For a geometric series, it’s $S_n = a_1 \frac{1 - r^n}{1 - r}$.

4. Application to Word Problems: Turn word problems into math problems by finding important numbers and how they relate to each other. For instance, if a word problem asks for the sum of the first $n$ terms of an arithmetic sequence, use the formulas mentioned earlier.

5. Tool Use: Use charts or tables to see how the sequence develops. This can really help in spotting the pattern.

These methods improve problem-solving skills and help students understand sequences and series better in math.