Number Sequences and nth Term
Learn about arithmetic and geometric sequences, how to find the nth term formula, Fibonacci numbers, and how to use sequences to spot patterns and predict values.
Sequences are lists of numbers that follow a pattern or rule. Once you spot the pattern, you can predict any number in the sequence — even the 100th or 1,000th.
Arithmetic Sequences (Adding the Same Amount)
An arithmetic sequence adds the same number each time. That number is called the common difference (d).
Example: 3, 7, 11, 15, 19, … → common difference = 4
Finding the nth Term
Where d is the common difference and a is the first term.
Worked Example 1 — Finding the nth Term
Sequence: 5, 8, 11, 14, …
- Common difference d = 3
- First term a = 5
- nth term = 3n + (5 − 3) = 3n + 2
- Check: 1st term = 3(1) + 2 = 5 ✓, 2nd = 3(2) + 2 = 8 ✓
Worked Example 2 — Finding the 50th Term
Using the sequence above (nth term = 3n + 2):
50th term = 3(50) + 2 = 152
Geometric Sequences (Multiplying by the Same Amount)
A geometric sequence multiplies by the same number each time. That number is the common ratio (r).
Example: 2, 6, 18, 54, … → common ratio = 3 (each term × 3)
Special Sequences
| Sequence | Numbers | Rule |
|---|---|---|
| Square numbers | 1, 4, 9, 16, 25, … | n² |
| Triangular numbers | 1, 3, 6, 10, 15, … | n(n+1)/2 |
| Cube numbers | 1, 8, 27, 64, … | n³ |
| Fibonacci | 1, 1, 2, 3, 5, 8, 13, … | Add previous two terms |
Worked Example 3 — Is 100 in the Sequence?
The nth term of a sequence is 4n − 3. Is 100 in the sequence?
- Set 4n − 3 = 100
- 4n = 103
- n = 25.75
- n must be a whole number, so 100 is NOT in the sequence
Try It Yourself
- Find the nth term for: 7, 10, 13, 16, … (Answer: 3n + 4)
- Find the 20th term of: 2, 5, 8, 11, … (Answer: 59)
- What is the 10th triangular number? (Answer: 55)