Sequences
Types of Sequence
Arithmetic (Linear) Sequences
Each term is obtained by adding (or subtracting) a constant difference $d$.
$$a,; a+d,; a+2d,; a+3d,; \ldots$$
nth term formula: $T_n = a + (n-1)d$, or equivalently $T_n = dn + c$
Example: 3, 7, 11, 15, 19, …
Common difference $d = 4$. First term $a = 3$. $$T_n = 4n - 1 \quad (\text{check: } T_1 = 3\checkmark,; T_2 = 7\checkmark)$$
Finding $d$ and $c$ from $T_n = dn + c$:
- $d$ = constant difference between consecutive terms.
- $c = T_1 - d$ (or match term values to find $c$).
Geometric Sequences
Each term is obtained by multiplying by a constant ratio $r$.
$$a,; ar,; ar^2,; ar^3,; \ldots$$
nth term: $T_n = a \cdot r^{n-1}$
Example: 2, 6, 18, 54, … (ratio $r = 3$; $T_n = 2 \times 3^{n-1}$)
Fibonacci Sequences
Each term is the sum of the two preceding terms.
$$1,; 1,; 2,; 3,; 5,; 8,; 13,; 21,; \ldots$$
$$F_n = F_{n-1} + F_{n-2}$$
Quadratic Sequences
Second differences are constant (but first differences are not).
| Position | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Term | 1 | 4 | 9 | 16 | 25 |
| 1st diff | 3 | 5 | 7 | 9 | |
| 2nd diff | 2 | 2 | 2 |
$T_n = n^2$ here. In general: if second difference $= 2a$, then the $n^2$ coefficient is $a$.
Example: 3, 8, 15, 24, 35, … (1st diff: 5, 7, 9, 11; 2nd diff: 2, 2, 2 → coefficient of $n^2$ is 1)
$T_n = n^2 + \ldots$; compare with $n^2 = 1, 4, 9, 16, 25$; differences are 2, 4, 6, 8, 10 → add $2n$; then adjust constant: $T_1 = 1 + 2 = 3$, so constant $= 0$. Hence $T_n = n^2 + 2n$.
Key Vocabulary
- Term-to-term rule: how to get from one term to the next (add $d$, multiply by $r$, etc.).
- Position-to-term rule (nth term): a formula giving the value of the $n$th term directly.
WJEC Exam Tips
- For arithmetic sequences: find $d$ from consecutive differences; $T_n = dn + c$.
- Always verify: substitute $n=1, n=2$ into your nth term formula.
- WJEC questions often ask: "Is 100 a term in this sequence?" — solve $T_n = 100$ and check if $n$ is a positive integer.
- For quadratic: second difference $= 2a$ where $a$ is the coefficient of $n^2$.
- Show all working when finding the nth term.
AI-generated · claude-opus-4-7 · v3-wjec-maths