Mini-Lesson
Sequences and series
This mini-lesson covers sigma notation and recurrence relations , arithmetic and geometric progressions with their sums, the sum to infinity when |r| < 1, and the binomial expansion — first for positive integer n, then for negative and fractional indices where validity |x| < 1 matters.
Where this sits: Unit AS 1: Pure Mathematics — sequences and series, arithmetic and geometric progressions, binomial expansion for positive integer index. Unit A2 1 — binomial expansion for any rational index, with the condition for validity.
Work through each screen, answer the questions as you go (some are reasoning, some are calculations) and collect ⭐ stars. Press Start when you are ready.
Notation
Sigma notation and recurrence
Σ (r = 1 to n) uᵣ = u₁ + u₂ + … + uₙΣ tells you to ADD the terms; the limits say where to start and stop
A sequence can be given by a position-to-term rule (uₙ = 3n + 2) or a recurrence relation (uₙ₊₁ = 2uₙ − 1 with u₁ = 3), which needs a starting value.
Worked example — recurrence
u₁ = 3, uₙ₊₁ = 2uₙ − 1
u₂ = 2(3) − 1 = 5 · u₃ = 2(5) − 1 = 9 · u₄ = 2(9) − 1 = 17
Sequences can be increasing (uₙ₊₁ > uₙ for all n), decreasing , or periodic (they repeat with period k, so uₙ₊ₖ = uₙ).
Arithmetic
Arithmetic progressions
uₙ = a + (n − 1)d · Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)a = first term, d = common difference, l = last term
Worked example — a = 5, d = 3
u₂₀ = 5 + 19 × 3 = 5 + 57 = 62
S₂₀ = 20/2 [2(5) + 19(3)] = 10[10 + 57] = 10 × 67 = 670
Check with the other formula: S₂₀ = 20/2 (5 + 62) = 10 × 67 = 670 ✓
Calculate
Arithmetic term
1 An arithmetic sequence has first term a = 5 and common difference d = 3. Find the 20th term .
Check ✓
Hint: uₙ = a + (n − 1)d = 5 + 19 × 3. Note it is (n − 1), not n.
Calculate
Arithmetic sum
2 For the same sequence (a = 5, d = 3), find the sum of the first 20 terms , S₂₀.
Check ✓
Hint: Sₙ = n/2 [2a + (n − 1)d] = 10 × [10 + 57] = 10 × 67. Or use S = n/2 (a + l) = 10 × (5 + 62).
Quick check
Which is arithmetic?
? Which sequence is arithmetic?
2, 6, 18, 54, … ❌
1, 4, 9, 16, … ❌
20, 15, 10, 5, … ✅
1, 1, 2, 3, 5, … ❌
Geometric
Geometric progressions and the sum to infinity
uₙ = arⁿ⁻¹ · Sₙ = a(1 − rⁿ)/(1 − r) · S∞ = a/(1 − r), valid only for |r| < 1r = common ratio = any term ÷ the term before it
Worked example — a = 12, r = 1/3
The terms 12, 4, 4/3, … shrink, and |r| = 1/3 < 1, so the sum to infinity exists.
S∞ = 12/(1 − 1/3) = 12/(2/3) = 12 × 3/2 = 18
Convergence: if |r| ≥ 1 the terms do not shrink to zero and there is no sum to infinity. Always state the condition |r| < 1 — it is a marked point.
Calculate
Sum to infinity
3 A geometric series has first term 12 and common ratio 1/3. Find the sum to infinity .
Check ✓
Hint: S∞ = a/(1 − r) = 12 ÷ (1 − 1/3) = 12 ÷ (2/3) = 12 × 3/2.
Sort it
Arithmetic, geometric or neither?
Tap a sequence, then tap the family it belongs to.
Binomial
The binomial expansion (positive integer n)
(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …or (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ — a FINITE expansion with n + 1 terms
Worked example — coefficient of x³ in (1 + 2x)⁶
Term = ⁶C₃ × (2x)³ = 20 × 8x³ = 160x³
(⁶C₃ = 6!/(3!3!) = 720/36 = 20, and 2³ = 8.)
Do not forget to cube the 2. Forgetting to raise the coefficient inside the bracket to the power is the single most common binomial error.
Calculate
Binomial coefficient
4 Find the coefficient of x³ in the expansion of (1 + 2x)⁶.
Check ✓
Hint: Coefficient = ⁶C₃ × 2³ = 20 × 8.
A2 binomial
Negative and fractional indices
For any rational n, the series continues for ever and is valid only when |x| < 1 :
(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + …, |x| < 1for (1 + bx)ⁿ the condition becomes |bx| < 1, i.e. |x| < 1/|b|
Worked example — (1 − 3x)^(1/2)
With n = ½ and "x" replaced by (−3x):
1 + ½(−3x) + [½(−½)/2](−3x)² + … = 1 − 1.5x + (−1/8)(9x²) + …
= 1 − 1.5x − 1.125 x² + …, valid for |3x| < 1, i.e. |x| < 1/3
Calculate
Fractional index expansion
5 Expand (1 − 3x)^(1/2) in ascending powers of x. State the coefficient of x² .
Check ✓
Hint: n = ½. The x² term is [n(n − 1)/2!](−3x)² = [½ × (−½) ÷ 2] × 9x² = (−1/8) × 9x² = −1.125x².
Quick check
Validity
? For what values of x is the expansion of (1 + 4x)⁻¹ valid?
all real x ❌
|x| < 1 ❌
|x| < 1/4 ✅
x > 0 ❌
Match it
Match the series result
Tap the item on the left, then its value on the right.
Quick check
Sum to infinity exists?
? Which geometric series has a sum to infinity?
first term 5, ratio 1.2 ❌
first term 5, ratio −0.8 ✅
first term 5, ratio 1 ❌
first term 5, ratio −2 ❌
Quick check
Sigma limits
? Σ (r = 3 to 7) (2r + 1) has how many terms?
4 ❌
5 ✅
7 ❌
10 ❌
Examiner traps
Sequences and binomial pitfalls
(n − 1)d, not nd. u₂₀ = a + 19d.S∞ without checking |r| < 1. State the condition — it carries a mark.Forgetting to raise the whole term to the power: in (1 + 2x)⁶ the x³ coefficient is ⁶C₃ × 2³ = 160, not 20.Validity of a rational-index expansion: (1 + bx)ⁿ needs |bx| < 1, i.e. |x| < 1/|b| — not |x| < 1.Number of terms in a sigma: upper − lower + 1.
Recap
The big ideas to know
AP: uₙ = a + (n − 1)d · Sₙ = n/2[2a + (n − 1)d]
GP: uₙ = arⁿ⁻¹ · Sₙ = a(1 − rⁿ)/(1 − r) · S∞ = a/(1 − r) only if |r| < 1
Recurrence: needs a starting term; sequences can be increasing, decreasing or periodic
Binomial (n ∈ ℕ): finite, use ⁿCᵣ — remember to raise the whole bracket term to the power
Binomial (rational n): infinite series, valid only for |x| < 1 (or |bx| < 1)
Binomial expansions with |x| < 1 reappear in A2 integration and approximation work. Press Finish to see your score.
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