AQA A-Level Mathematics: D: Sequences and series (7357) – Revision Notes

These notes condense Section D: Sequences and series (D1 to D6) of AQA A-Level Mathematics (7357), from the AQA A-level Mathematics specification, version 1.3 (31 January 2018), for exams from June 2018 onwards. The specification lists Section D under Paper 1, and Papers 2 and 3 can assess any Paper 1 content. For full explanations and worked examples, read the Section D study guide. To practise exam-style questions, use the Section D practice set.

Course links: AQA A-Level Mathematics hub and printable 7357 checklist.

Formulas at a glance

Result	Formula	Condition
Factorial	n! = n(n − 1)…2 × 1, 0! = 1	n a non-negative integer
Binomial coefficient	ⁿCᵣ = n!/(r!(n − r)!)	n, r integers, 0 ≤ r ≤ n
Coefficient, rational n	n(n − 1)…(n − r + 1)/r!	any rational n
Binomial, positive integer n	(a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ, r = 0 to n	exact, n + 1 terms
Binomial, rational n	(1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + …	|x| < 1
General form	(a + bx)ⁿ = aⁿ(1 + bx/a)ⁿ	|bx/a| < 1
Arithmetic nth term	uₙ = a + (n − 1)d	recall (Appendix B)
Arithmetic sum	Sₙ = (n/2)(2a + (n − 1)d) = (n/2)(a + l)
Geometric nth term	uₙ = arⁿ⁻¹	recall (Appendix B)
Geometric sum	Sₙ = a(1 − rⁿ)/(1 − r)	r ≠ 1
Sum to infinity	S∞ = a/(1 − r)	|r| < 1 only
Useful sum	Σ r (r = 1 to n) = n(n + 1)/2	it is arithmetic

Notation from the specification: a first term, l last term, d common difference, r common ratio, Sₙ sum to n terms, S∞ sum to infinity.

D1 – Binomial expansion

Method: positive integer n

1. Write the general term ⁿCᵣ aⁿ⁻ʳ (bx)ʳ.
2. Bracket bx completely, sign included.
3. For a single coefficient, pick the r that gives the power you need. Don’t expand everything.

Method: rational n

1. Rewrite as aⁿ(1 + (b/a)x)ⁿ. For a fractional n, aⁿ may be a root: 8^(1/3) = 2.
2. Use 1 + nX + n(n − 1)X²/2! + … with X = (b/a)x, bracketing X each time.
3. Multiply every term by aⁿ.
4. State validity: |bx/a| < 1, or equivalently |x| < |a/b|.

Approximation: pick x small and inside the valid range so the ignored terms are tiny. Check the substitution gives the number asked for.

Link to probability: for X ~ B(n, p), P(X = r) = ⁿCᵣ pʳ(1 − p)ⁿ⁻ʳ is a term of ((1 − p) + p)ⁿ.

Worked reminders:

• (1 − x)⁻¹ = 1 + x + x² + x³ + …, valid for |x| < 1.
• (2 + x)⁻¹ = (1/2)(1 + x/2)⁻¹ = (1/2)(1 − x/2 + x²/4 − …) = 1/2 − x/4 + x²/8 − …, valid for |x| < 2. Taking out 2⁻¹ = 1/2 first is the step that is most often skipped.

D2 – Sequences

• nth-term formula: uₙ given directly; substitute n.
• Recurrence: xₙ₊₁ = f(xₙ) plus a first term; generate one term at a time.
• Increasing: uₙ₊₁ − uₙ > 0 for all n. Decreasing: uₙ₊₁ − uₙ < 0 for all n.
• Periodic: terms repeat after k steps, k the order. Find a distant term by the remainder when its position is divided by k.
• Tip: rewrite a fraction like (2n + 3)/(n + 1) as 2 + 1/(n + 1) to see its behaviour at a glance.

Worked reminder: for uₙ = (2n + 3)/(n + 1),

uₙ₊₁ − uₙ = 1/(n + 2) − 1/(n + 1) = −1/((n + 1)(n + 2)) < 0

so the sequence is decreasing for every n ≥ 1, and the terms approach 2 from above.

D3 – Sigma notation

• Σ from r = k to m of f(r): substitute each integer from k to m and add.
• Number of terms = m − k + 1.
• Σ c (a constant) over N terms = cN.
• Σ (f(r) + g(r)) = Σ f(r) + Σ g(r), and Σ c f(r) = c Σ f(r).
• Sum from r = k to m = (sum from 1 to m) − (sum from 1 to k − 1).

Worked reminder: Σ from r = 5 to 12 of r² has 12 − 5 + 1 = 8 terms: 25 + 36 + 49 + 64 + 81 + 100 + 121 + 144 = 620. Writing out the first and last terms is the quickest way to confirm the count.

D4 and D5 – Method in steps

Finding a and d (or a and r) from two terms

1. Write each given term with the nth-term formula.
2. Arithmetic: subtract the equations. Geometric: divide them.
3. Back-substitute for a.

Least n so that a sum passes a target

1. Write the inequality with Sₙ.
2. Arithmetic: rearrange to a quadratic in n, solve, round up, then check both neighbours.
3. Geometric: isolate rⁿ, take logs. If 0 < r < 1, dividing by ln r flips the inequality sign.

Worked reminders:

• Arithmetic, a = 50, d = −4. Positive terms need 50 − 4(n − 1) > 0, so n < 13.5 and there are 13 positive terms (u₁₃ = 2, u₁₄ = −2). Their sum is (13/2)(50 + 2) = 338.
• Geometric, a = 5, r = 3. S₆ = 5(3⁶ − 1)/(3 − 1) = 5 × 728/2 = 1820. When r > 1, the form a(rⁿ − 1)/(r − 1) avoids negative signs.

Convergence of a series in x

1. Identify r as an expression in x.
2. Solve |r| < 1, that is −1 < r < 1.
3. Give the range of x; only then use S∞.

D6 – Modelling

Situation	Model
Fixed amount added each step	Arithmetic
Fixed percentage change each step	Geometric, r = 1 ± rate
Repeated dose with a fixed fraction lost	Recurrence xₙ₊₁ = kxₙ + c
Total over many steps	The matching series

Worked reminder: a ball dropped from 2 m rebounds to 60% of its previous height each time. The rebound heights 1.2, 0.72, … form a geometric series with S∞ = 1.2/(1 − 0.6) = 3. Each rebound is travelled up and down, so the total distance is 2 + 2 × 3 = 8 m. The model predicts infinitely many bounces, which is its obvious limitation.

Always say what n counts and whether term 1 is “now” or “after one step”. Name one limitation: growth cannot continue forever, rates change, money is in whole pence. For modelling as an overarching theme, see the overarching themes guide.

Must-know distinctions

• Arithmetic vs geometric: constant difference vs constant ratio. Test with u₂ − u₁ = u₃ − u₂, or u₂/u₁ = u₃/u₂.
• uₙ vs Sₙ: a single term vs a running total. Sₙ − Sₙ₋₁ = uₙ.
• Finite vs infinite expansion: positive integer n stops; negative or fractional n does not and needs a validity condition.
• Convergent sequence vs convergent series: for a geometric sequence with |r| < 1 the terms tend to 0 and the series tends to a/(1 − r).
• Increasing vs periodic: a periodic sequence is neither increasing nor decreasing.

Quick self-test

1. Evaluate ⁶C₂.
2. Simplify 5!/3!.
3. Expand (1 + x)⁻¹ up to x² and state when it is valid.
4. Find the coefficient of x² in (1 − 2x)^(1/2).
5. Evaluate Σ from r = 1 to 10 of 2ʳ.
6. An arithmetic sequence has a = 4 and d = −3. Find u₁₅.
7. Find 1 + 2 + 3 + … + 100.
8. Find S∞ for a geometric series with a = 10 and r = −0.6.
9. u₁ = 4 and uₙ₊₁ = −uₙ. Describe the sequence.
10. Does 3 + 6 + 12 + … have a sum to infinity? Why?
11. x₁ = 2 and xₙ₊₁ = xₙ/2 + 3. Find x₂ and x₃.
12. For which x is the expansion of (3 − 2x)⁻¹ valid?

Answers

1. 15
2. 20
3. 1 − x + x², valid for |x| < 1
4. −1/2
5. 2046
6. −38
7. 5050
8. 6.25
9. 4, −4, 4, −4, …: periodic with order 2
10. No: r = 2 and |r| ≥ 1, so the series diverges
11. x₂ = 4, x₃ = 5
12. |x| < 3/2

Where marks are usually lost

• Writing (2x)³ as 2x³ inside a binomial term, which makes that coefficient wrong and loses the accuracy mark.
• Using (1 + x)ⁿ for (4 − x)^(1/2) without first taking out 4^(1/2) = 2.
• Giving the validity of (a + bx)ⁿ as |x| < 1 when it should be |x| < |a/b|.
• Rounding a “least n” answer down instead of up, or failing to check the neighbouring integer.
• Not flipping the inequality when dividing by ln r with 0 < r < 1.
• Using S∞ when |r| ≥ 1; the question usually expects you to say no sum exists.
• Miscounting terms in sigma notation that does not start at r = 1.
• Losing the minus sign in r when solving r³ = negative number.
• In models, giving an answer with no context or units, or calling a geometric model “linear”.

Next steps

Work through the practice questions, revisit the study guide for any gap, and check paper timing with the 7357 exam preparation guide. Find weak topics fast with the free diagnostics.

Official syllabus

AQA A-level Mathematics (7357) specification, version 1.3, 31 January 2018, for teaching from September 2017 and exams from June 2018 onwards, published by AQA. Section 3.5, D: Sequences and series (D1 to D6).