1. Chapter at a glance
- A sequence is an ordered list of numbers (finite or infinite) in which each number is a term; notation uses t₁, t₂, … (or sₙ, uₙ) with subscript matching term position.
- An explicit rule expresses the nth term directly as a formula in n (e.g., uₙ = 2n – 1 for odd numbers).
- A recursive rule defines each term from one or more previous terms (e.g., t₁ = a, tₙ = tₙ₋₁ + d for n ≥ 2).
- An arithmetic progression (AP) is a sequence in which the difference between consecutive terms is constant (common difference d); general form a, a + d, a + 2d, … .
- The sum of the first n natural numbers is Sₙ = n(n + 1)/2, derived by pairing the sum with its reverse.
- A geometric progression (GP) is a sequence in which each term after the first is obtained by multiplying the previous term by a constant (common ratio r); general form a, ar, ar², … .
- Both AP and GP admit explicit and recursive rules; their graphs (stage number vs. term) are linear for AP and non-linear for GP.
- Patterns such as triangular numbers, square numbers and certain fractals (Sierpiński triangle) are special sequences or progressions.
2. Definitions, theorems and results
- Sequence: an ordered list of numbers; each number is a term of the sequence. Sequences may be finite or infinite.
- Explicit rule (formula): a rule that uses the term’s position number n to calculate its value directly.
- Recursive rule (formula): a rule that gives the value of a term using the values of previous term(s); earlier term(s) must be known.
- Arithmetic progression (AP): a sequence in which the difference between any two consecutive terms is constant (called the common difference d). General form: a, a + d, a + 2d, …, a + (n – 1)d.
- nth term of an AP: tₙ = a + (n – 1)d (explicit); recursive form t₁ = a, tₙ = tₙ₋₁ + d (n ≥ 2).
- Sum of first n natural numbers: Sₙ = n(n + 1)/2 (derived by writing S = 1 + 2 + … + n and its reverse, adding to obtain 2S = n(n + 1)).
- Geometric progression (GP): a sequence in which each term after the first is obtained by multiplying the previous term by a fixed number (common ratio r). General form: a, ar, ar², …, ar^{n–1}.
- nth term of a GP: tₙ = a r^{n–1} (explicit); recursive form t₁ = a, tₙ = r tₙ₋₁ (n ≥ 2).
No theorems, lemmas or axioms requiring formal proof are stated in the chapter.
3. Formula sheet
| Formula |
Meaning of symbols |
| tₙ = f(n) |
explicit rule; tₙ is nth term, n = position (natural number) |
| t₁ = given, tₙ = expression involving tₙ₋₁ (or tₙ₋₂) |
recursive rule |
| tₙ = a + (n – 1)d |
AP nth term; a = first term, d = common difference |
| t₁ = a, tₙ = tₙ₋₁ + d (n ≥ 2) |
AP recursive rule |
| Sₙ = n(n + 1)/2 |
sum of first n natural numbers |
| tₙ = a r^{n–1} |
GP nth term; a = first term, r = common ratio |
| t₁ = a, tₙ = r tₙ₋₁ (n ≥ 2) |
GP recursive rule |
4. Solved-example patterns
- Find first k terms or a specific term given explicit rule: substitute successive natural numbers for n (or the required n) into the formula.
- Find first k terms given recursive rule: start with the given first term(s) and repeatedly apply the recurrence relation.
- Check whether a number belongs to a sequence and find its position: solve the equation tₙ = given number for n; verify that the solution is a natural number.
- Identify whether a sequence is an AP or GP and find its nth term: compute successive differences (for AP) or ratios (for GP); if constant, write the general term using a and d (or r).
- Find the sum of first n natural numbers or difference of two such sums: apply Sₙ = n(n + 1)/2 directly or as S_b – S_a for consecutive sums.
- Solve word problems involving AP/GP (fare, height, bacteria, etc.): translate the constant increment or multiplier into d or r, then apply the appropriate explicit or recursive formula.
5. Common mistakes and exam pitfalls
- Treating a non-natural solution for n as valid (e.g., n = 94.6) when checking membership of a sequence.
- Confusing the recursive rule with the explicit rule and attempting to compute later terms without the initial term(s).
- Forgetting that the common difference d in an AP may be negative (decreasing sequences).
- Using the sum formula Sₙ = n(n + 1)/2 for a sequence that is not the natural numbers.
- Writing the GP nth term as ar^n instead of ar^{n–1}.
- Omitting the condition “n ≥ 2” (or n ≥ 3) when stating a recursive rule.
- Plotting stage-number vs. term and expecting a straight line for a GP.