Identifying the Pattern

Find the pattern and the next three terms: $3, 7, 11, 15, \ldots$

Find the difference between consecutive terms: $7 - 3 = 4$, $11 - 7 = 4$, $15 - 11 = 4$ = Common difference = 4

Identify the pattern: Each term is 4 more than the previous term = Add 4 to get next term

Find the next three terms: $15 + 4 = 19$, $19 + 4 = 23$, $23 + 4 = 27$ = $19, 23, 27$

Answer: Pattern: Add 4. Next three terms: $19, 23, 27$

Writing a Formula for the nth Term

Write a formula for the $n$th term of the sequence: $5, 8, 11, 14, \ldots$

Identify the first term and common difference: $a_1 = 5$, $d = 8 - 5 = 3$ = First term = 5, difference = 3

Use the arithmetic sequence formula: $a_n = a_1 + (n-1)d$ = $a_n = 5 + (n-1) \times 3$

Simplify the formula: $a_n = 5 + 3n - 3 = 3n + 2$ = $a_n = 3n + 2$

Verify with known terms: $a_1 = 3(1) + 2 = 5$ \checkmark, $a_2 = 3(2) + 2 = 8$ \checkmark = Formula verified!

Answer: $a_n = 3n + 2$

Finding a Specific Term

In the sequence $2, 6, 18, 54, \ldots$, find the 7th term.

Identify the pattern: $\frac{6}{2} = 3$, $\frac{18}{6} = 3$, $\frac{54}{18} = 3$ = Common ratio = 3 (multiply by 3)

Write the formula for geometric sequence: $a_n = a_1 \times r^{n-1}$ where $a_1 = 2$ and $r = 3$ = $a_n = 2 \times 3^{n-1}$

Substitute n = 7: $a_7 = 2 \times 3^{7-1} = 2 \times 3^6$ = $a_7 = 2 \times 729$

Calculate: $a_7 = 1458$ = The 7th term is 1458

Answer: $a_7 = 1458$

Mistake: Confusing the term number with the term value

Why: In $a_5 = 17$, students mix up 5 (position) with 17 (value).

Correct: $a_n$ means the value of the term at position $n$. If $a_5 = 17$, then the 5th term has a value of 17.

Mistake: Using $n$ instead of $(n-1)$ in the formula

Why: The formula $a_n = a_1 + nd$ gives wrong answers because it counts the difference one extra time.

Correct: Use $a_n = a_1 + (n-1)d$ because you add the difference $(n-1)$ times to reach the $n$th term.

Mistake: Assuming all sequences are arithmetic

Why: Not all sequences have a constant difference. Some multiply (geometric) or follow other patterns.

Correct: Always check both differences AND ratios between terms to identify the type of sequence.

Saving Money

If you save a fixed amount each month, your total savings forms an arithmetic sequence.

Save 100 euros per month: 100, 200, 300, 400, ... After $n$ months: $a_n = 100n$ euros.

Bacterial Growth

Bacteria often double every hour, creating a geometric sequence.

Starting with 100 bacteria that double hourly: 100, 200, 400, 800, ... After $n$ hours: $a_n = 100 \times 2^{n-1}$

Q: What is the difference between a sequence and a series?

A: A sequence is an ordered list of numbers (e.g., 2, 4, 6, 8). A series is the SUM of a sequence's terms (e.g., 2 + 4 + 6 + 8 = 20).

Q: Can a sequence have negative terms?

A: Yes! For example, $5, 2, -1, -4, \ldots$ is an arithmetic sequence with common difference $-3$.

Q: Why do we use $(n-1)$ in the formulas?

A: Because the first term doesn't need any additions or multiplications. To get to $a_2$, we add once; to $a_3$, twice; to $a_n$, we add $(n-1)$ times.