Intro
Sequence and series questions reward formula recognition: identify arithmetic vs geometric, write the nth term, then apply sum formulas or sigma notation carefully. Show substitution into Sn = n/2(2a + (n−1)d) or Sn = a(1 − rn)/(1 − r) — method marks matter. For NSW HSC Mathematics Extension 1. Keywords: arithmetic and geometric series HSC, sigma notation help, sequence revision.
Summary
Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio. Sigma notation requires careful index shifts. Watch for |r| = 1, off-by-one term counts, and whether the question wants exact or decimal form.
Series questions sometimes embed within financial or growth contexts — identify a, d, or r from the wording before choosing a formula. Show clear working when solving for n in index positions.
Key Points
- Arithmetic: Tn = a + (n−1)d; sum Sn = n/2(2a + (n−1)d).
- Geometric: Tn = arn−1; sum Sn = a(1 − rn)/(1 − r) for r ≠ 1.
- Infinite GP sum S∞ = a/(1 − r) only when |r| < 1.
- Recurrence relations: check if Tn − Tn−1 is constant (AP) or Tn/Tn−1 is constant (GP).
- Sigma rules: split sums and factor constants — watch index boundaries.
- Work through HSC Sequences booklet sigma examples.
Worked example
Question. The first term of a GP is 3 and the common ratio is 2. Find the sum of the first 8 terms.
Solution.
- Identify GP: a = 3, r = 2, n = 8.
- S8 = a(1 − r8)/(1 − r) = 3(1 − 256)/(1 − 2).
- = 3(−255)/(−1) = 765.
Answer. 765.
Takeaway. State a, r, and n before substituting — GP sum formula fails visibly when r = 1, so note that edge case.
Exam Preparation
Sequences are high-yield for Extension 1: clear formulas and careful algebra. Build speed by doing mixed AP/GP recognition drills, then past-paper sections under time. Log sigma index errors separately — they recur.
Infinite geometric series questions often appear as limiting values in growth contexts — state the condition |r| < 1 explicitly before applying S∞ = a/(1 − r). For finite sums, confirm the number of terms from the wording ('first eight terms' means n = 8, not 7).
- Formula sheet from memory. Write AP and GP nth term and sum formulas without notes.
- Sigma practice. Rewrite sums with shifted indices; verify with small n.
- Mixed past questions. Complete one Extension 1 sequences section per fortnight in Term 3.
Financial mathematics and recurrence word problems often hide AP or GP structure — write the first three terms before naming the sequence type. Sigma notation questions may split a sum into two parts with different indices; re-index carefully and verify with n = 1, 2, 3. Extension 1 trial papers regularly include simultaneous recurrence relations; express both sequences and eliminate one variable. Show substitution into sum formulas line-by-line for method marks even when arithmetic is routine.
Mini-FAQ
How do I tell AP from GP quickly?
Compute T2 − T1 and T3 − T2 — constant difference means AP. Compute ratios T2/T1, T3/T2 — constant ratio means GP.
When can I use the infinite GP formula?
Only when |r| < 1. If |r| ≥ 1, the series diverges.
Do recurrence relations always mean AP or GP?
No — but many HSC questions reduce to AP or GP after one or two terms. Write out T1, T2, T3 to spot the pattern.
Common mistakes to avoid
- Using GP sum with r = 1 (formula divides by zero).
- Off-by-one: confusing the nth term index with the number of terms.
- Applying infinite sum formula when the question asks for finitely many terms.
- Rounding too early in multi-step sigma problems.
Practice on Vu's Maths Hub
Need more practice on this topic? Open the free HSC Sequences booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required.
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