Sequences and Series
Patterns in numbers — from arithmetic and geometric progressions to the elegant relationship between AM and GM
In this chapter
8.1 Sequences and Series
"Natural numbers are the product of human spirit." — Dedekind
A sequence is an ordered list of numbers following a specific rule. Think of it as a function whose domain is natural numbers. Examples: 2, 4, 8, 16, ... (ancestors) or 3, 3.3, 3.33, 3.333, ... (successive quotients).
Types of Sequences
Finite Sequence
Has a fixed number of terms. Example: 2, 4, 8, 16 (4 terms)
Infinite Sequence
Goes on forever. Example: 2, 4, 6, 8, ... (never ends)
General Term
The general term (aₙ) gives the formula for any term:
• Even numbers: aₙ = 2n → 2, 4, 6, 8, ...
• Odd numbers: aₙ = 2n - 1 → 1, 3, 5, 7, ...
• Powers of 2: aₙ = 2ⁿ → 2, 4, 8, 16, ...
Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, ...
Rule: Each term = sum of previous two terms
a₁ = a₂ = 1, and aₙ = aₙ₋₁ + aₙ₋₂ for n > 2
Found in nature: sunflower spirals, rabbit populations, pinecones!
Series
A series is the sum of terms of a sequence. Written using sigma notation:
8.2 Geometric Progression (GP)
A Geometric Progression is a sequence where each term is obtained by multiplying the previous term by a constant ratio (r).
Key property: aₖ₊₁ / aₖ = r (constant) for all k ≥ 1
Examples:
• 2, 4, 8, 16, ... → r = 2 (multiply by 2 each time)
• 1, -1/3, 1/9, -1/27, ... → r = -1/3
• 100, 10, 1, 0.1, ... → r = 0.1
Standard Form
Where a = first term, r = common ratio
To check if a sequence is GP:
Divide each term by the previous one. If the ratio is constant, it's a GP.
Example: 3, 6, 12, 24, ...
6/3 = 2, 12/6 = 2, 24/12 = 2 → r = 2 → It's a GP!
8.3 General Term of GP
The nth term of a GP with first term a and common ratio r:
a = 5, r = 5
a₁₀ = 5 × 5¹⁰⁻¹ = 5 × 5⁹ = 5¹⁰ = 9,765,625
GP: 2, 8, 32, ... → a = 2, r = 4
131072 = 2 × 4ⁿ⁻¹
65536 = 4ⁿ⁻¹
4⁸ = 4ⁿ⁻¹ → n - 1 = 8 → n = 9
So 131072 is the 9th term.
If 3rd term = 24 and 6th term = 192:
ar² = 24 and ar⁵ = 192
Divide: r³ = 192/24 = 8 → r = 2
Substitute: a(2²) = 24 → a = 6
8.4 Sum of GP
Sum of First n Terms
When r ≠ 1:
Sₙ = a(rⁿ - 1) / (r - 1) or Sₙ = a(1 - rⁿ) / (1 - r)
When r = 1:
Sₙ = a + a + a + ... (n times) = na
GP: 1, 2/3, 4/9, ... → a = 1, r = 2/3
S₅ = 1 × (1 - (2/3)⁵) / (1 - 2/3)
= (1 - 32/243) / (1/3)
= (211/243) × 3 = 211/81
Sum of Infinite GP
S∞ = a / (1 - r)
This works only when |r| < 1 (the terms get smaller and smaller).
Example: 1, 1/2, 1/4, 1/8, ...
S∞ = 1 / (1 - 1/2) = 1 / (1/2) = 2
0.999... = 0.9 + 0.09 + 0.009 + ...
This is a GP with a = 0.9, r = 0.1
S∞ = 0.9 / (1 - 0.1) = 0.9 / 0.9 = 1
So yes, 0.999... = 1 exactly!
8.5 Geometric Mean (GM)
The geometric mean of two positive numbers a and b is:
GM of 2 and 8 = √(2 × 8) = √16 = 4
Notice: 2, 4, 8 are in GP! (4/2 = 2, 8/4 = 2)
Inserting GMs Between Two Numbers
We need: 1, G₁, G₂, G₃, 256 to be in GP
256 = 1 × r⁴ → r⁴ = 256 → r = ±4
If r = 4: 1, 4, 16, 64, 256
If r = -4: 1, -4, 16, -64, 256
8.6 Relationship Between AM and GM
For any two positive numbers a and b:
Arithmetic Mean (AM)
AM = (a + b) / 2
Geometric Mean (GM)
GM = √(ab)
For any two positive numbers: AM ≥ GM
Proof: AM - GM = (a + b)/2 - √(ab)
= (a + b - 2√(ab)) / 2
= (√a - √b)² / 2 ≥ 0
Since (√a - √b)² is always ≥ 0, AM ≥ GM.
Equality holds only when a = b.
If AM = 10 and GM = 8, find the numbers:
(a + b)/2 = 10 → a + b = 20
√(ab) = 8 → ab = 64
Solving: a = 4, b = 16 (or vice versa)
8.7 Summary
• Sequence: Ordered list of numbers following a rule. Series: Sum of sequence terms.
• GP: Each term = previous term × r. General term: aₙ = arⁿ⁻¹
• Sum of GP (r≠1): Sₙ = a(rⁿ - 1)/(r - 1) or a(1 - rⁿ)/(1 - r)
• Infinite GP (|r|<1): S∞ = a/(1-r)
• Geometric Mean: GM = √(ab)
• AM ≥ GM for positive numbers (equality when a = b)