Sequences and Series: AP and GP

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.

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.

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 + 16 = 23, so a = 7

For S₂₀: S₂₀ = 20/2[2(7) + 19(4)] = 10[14 + 76] = 900

Examiner Note: Show clear working for simultaneous equations (method marks available)

Example 2: A GP has first term 8 and common ratio 1.5. Find which term first exceeds 1000.

Solution: Need uₙ > 1000, so 8(1.5)ⁿ⁻¹ > 1000

(1.5)ⁿ⁻¹ > 125

Taking logs: (n-1)log(1.5) > log(125)

n-1 > log(125)/log(1.5) = 11.91...

n > 12.91, so n = 13 (first integer value)

Check: u₁₃ = 8(1.5)¹² ≈ 1034 ✓

Example 3: Find the sum to infinity of 12 + 9 + 6.75 + ...

Solution: r = 9/12 = 0.75, |r| < 1 ✓ converges

S∞ = a/(1-r) = 12/(1-0.75) = 12/0.25 = 48