Sequences and series

Sequences are ordered lists of numbers following a specific rule, denoted as {u₁, u₂, u₃, ...} or {uₙ}. Each number is a term, with uₙ representing the nth term.

Arithmetic Sequences have a constant difference between consecutive terms. The general term is: uₙ = u₁ + (n-1)d, where u₁ is the first term and d is the common difference. The sum of n terms (Sₙ) is: Sₙ = n/2[2u₁ + (n-1)d] or Sₙ = n/2(u₁ + uₙ).

Geometric Sequences have a constant ratio between consecutive terms. The general term is: uₙ = u₁r^(n-1), where r is the common ratio. The sum of n terms is: Sₙ = u₁(1-rⁿ)/(1-r) for r ≠ 1, or Sₙ = u₁(rⁿ-1)/(r-1).

Infinite Geometric Series converge when |r| < 1, with sum: S∞ = u₁/(1-r).

Sigma Notation (Σ) represents sums compactly: Σ(k=1 to n) uₖ means u₁ + u₂ + ... + uₙ.

Memory Aid - "APE": Arithmetic uses Plus (addition), Exponential (Geometric) uses times (multiplication).

Key formulas to memorize:

• Σ(k=1 to n) k = n(n+1)/2 (sum of first n natural numbers)
• Σ(k=1 to n) k² = n(n+1)(2n+1)/6
• Σ(k=1 to n) k³ = [n(n+1)/2]²

These foundational concepts underpin compound interest calculations, population models, and mathematical analysis at IB level.

Arithmetic sequences model linear growth scenarios. A savings plan where you deposit £50 monthly demonstrates this: Month 1 = £50, Month 2 = £100, Month 3 = £150. Here u₁ = 50, d = 50, so after 12 months: S₁₂ = 12/2[2(50) + 11(50)] = £3,900.

Geometric sequences represent exponential change. Bacterial growth where a population doubles every hour follows uₙ = u₁(2)^(n-1). Starting with 100 bacteria: after 5 hours = 100(2)⁴ = 1,600 bacteria. This models compound interest, radioactive decay, and viral spread.

Convergent series appear in fractals like Koch snowflake. Each iteration adds triangles with sides 1/3 the previous length. The perimeter forms a geometric series with r = 4/3 (divergent), while the area converges because subsequent additions decrease geometrically with |r| < 1.

Mortgage repayments combine both: if you borrow £200,000 at 3% annual interest with monthly payments, the outstanding balance forms a geometric sequence modified by arithmetic payments. The formula becomes: Remaining = P(1+r)ⁿ - PMT[(1+r)ⁿ - 1]/r.

Analogy: Arithmetic sequences are like climbing stairs with equal steps; geometric sequences are like folding paper—each fold doubles the thickness exponentially.

Medical dosage calculations use geometric decay: if a drug has a 6-hour half-life and you take 400mg, the amount remaining after n periods: uₙ = 400(0.5)^(n-1). Understanding convergence helps determine safe dosing intervals.

Example 1: An arithmetic sequence has u₃ = 17 and u₇ = 33. Find u₁, d, and S₂₀.

Solution: Using uₙ = u₁ + (n-1)d:

• u₃ = u₁ + 2d = 17 ... (1)
• u₇ = u₁ + 6d = 33 ... (2)

Subtract (1) from (2): 4d = 16, so d = 4

Substitute into (1): u₁ + 2(4) = 17, so u₁ = 9

For S₂₀: S₂₀ = 20/2[2(9) + 19(4)] = 10[18 + 76] = 940

Examiner note: Show clear algebraic steps. Marks are awarded for method even if arithmetic errors occur.

Example 2: A geometric sequence has u₂ = 6 and u₅ = 162. Find r, u₁, and determine if Σ(n=1 to ∞) uₙ converges.

Solution: Using uₙ = u₁r^(n-1):

• u₂ = u₁r = 6 ... (1)
• u₅ = u₁r⁴ = 162 ... (2)

Divide (2) by (1): r³ = 27, so r = 3

From (1): u₁(3) = 6, so u₁ = 2

Since |r| = 3 > 1, the series diverges (does not converge).

Example 3: Evaluate Σ(k=1 to 50) (3k - 2).

Solution: Split the sum: Σ(3k - 2) = 3Σk - Σ2 = 3[50(51)/2] - 2(50) = 3(1275) - 100 = 3,725