sequences series ap gp
Overview
# Sequences and Series: Arithmetic and Geometric Progressions This A-Level topic covers arithmetic progressions (AP) with constant differences and geometric progressions (GP) with constant ratios, including derivation and application of formulae for nth terms (aₙ = a + (n-1)d and aₙ = arⁿ⁻¹) and sum formulae. Students must master convergence conditions for infinite GP series (|r| < 1), solve real-world problems involving compound interest and population growth, and manipulate sigma notation—skills frequently examined through multi-step problems worth 6-9 marks. This foundation is essential for further pure mathematics topics including binomial expansion, calculus series, and mathematical modelling applications.
Core Concepts & Theory
Arithmetic Progressions (AP) are sequences where each term differs from the previous by a constant value called the common difference (d). The general term is given by: uₙ = a + (n-1)d, where a is the first term and n is the term number.
The sum of the first n terms of an AP is: Sₙ = n/2[2a + (n-1)d] or equivalently Sₙ = n/2(a + l), where l is the last term. These formulas are essential for Cambridge examinations and must be memorized.
Geometric Progressions (GP) are sequences where each term is obtained by multiplying the previous term by a constant called the common ratio (r). The general term is: uₙ = arⁿ⁻¹, where a is the first term.
The sum of the first n terms of a GP is: Sₙ = a(1-rⁿ)/(1-r) for r ≠ 1, or alternatively Sₙ = a(rⁿ-1)/(r-1). For an infinite GP where |r| < 1, the sum converges to: S∞ = a/(1-r).
Memory Aid (GRID): GP needs a Ratio, AP needs a Difference
Key Vocabulary: Sequence (ordered list of numbers), Series (sum of sequence terms), Convergent (approaches a finite limit), Divergent (grows without bound). Understanding these Cambridge-standard definitions ensures precise mathematical communication in examinations.
Detailed Explanation with Real-World Examples
Arithmetic Progressions model situations with constant change. Consider a salary increase: if you start at £25,000 with annual raises of £2,000, your salary forms an AP: 25000, 27000, 29000... Your salary in year 10 is u₁₀ = 25000 + 9(2000) = £43,000. Total earnings over 10 years: S₁₀ = 10/2[2(25000) + 9(2000)] = £340,000.
Geometric Progressions represent exponential growth or decay. Bacterial growth doubles every hour: starting with 100 bacteria, after 6 hours you have u₆ = 100(2)⁵ = 3,200 bacteria. Depreciation follows GP too: a £20,000 car losing 15% value annually has value u₅ = 20000(0.85)⁴ ≈ £10,470 after 4 years.
Financial applications are particularly relevant: compound interest follows GP formula A = P(1+r)ⁿ, while simple interest forms an AP. A £1,000 investment at 5% compound interest becomes 1000(1.05)¹⁰ ≈ £1,629 after 10 years.
Analogy: Think of AP as climbing stairs (equal steps), GP as a photocopy machine (each copy is a fixed scale of the previous)
The infinite GP sum applies to repeating decimals: 0.333... = 3/10 + 3/100 + 3/1000... = (3/10)/(1-1/10) = 1/3. This elegant connection demonstrates how infinite processes can yield finite results when |r| < 1, a concept Cambridge examiners frequently test.
Worked Examples & Step-by-Step Solutions
**Example 1:** The 5th term of an AP is 23 and the 12th term is 51. Find the first term and common difference, then calculate S₂₀. *Solution:* Using uₙ = a + (n-1)d: - u₅ = a + 4d = 23 ... (1) - u₁₂ = a + 11d = 51 ... (2) Subtracting (1) from (2): 7d = 28, so **d = 4** Substituting into (1): a + ...
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Key Concepts
- Sequence: An ordered list of numbers.
- Series: The sum of the terms of a sequence.
- Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant (common difference).
- Geometric Progression (GP): A sequence where the ratio of consecutive terms is constant (common ratio).
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Exam Tips
- →Always state the type of progression (AP or GP) and clearly write down the values of 'a', 'd'/'r' before applying formulas.
- →Be careful with calculations involving negative common ratios in GP, especially when raising them to powers.
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