Sequences and Series
Sequences are ordered lists following a pattern; series are sums of sequences.
Start with the simplest version: this lesson is about Sequences and Series. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.
Sequences are ordered lists following a pattern; series are sums of sequences. If you deposit money monthly with interest, your savings form a sequence growing over time, and the total represents a series. Understanding sequences and series is essential for modeling growth, decay, finance, and numerous natural phenomena. This chapter explores arithmetic progressions (where successive terms differ by a constant), geometric progressions (where ratios are constant), and their sums. These concepts form the foundation for calculus, infinite series, and applications throughout science and commerce.
Sequences: Ordered Lists with Pattern
A sequence is a function whose domain is natural numbers. We write it as {aₙ} or a₁, a₂, a₃, ..., where aₙ is the nth term.
Examples:
- {2, 4, 6, 8, 10, ...} is the sequence of positive even numbers
- {1, 1, 2, 3, 5, 8, ...} is the Fibonacci sequence, where each term is the sum of the previous two
Sequences can be finite (ending) or infinite (continuing forever).
Arithmetic Progressions (AP)
An arithmetic progression is a sequence where consecutive terms differ by a constant common difference d.
Example: 3, 7, 11, 15, 19, ... (d = 4)
General term: aₙ = a₁ + (n-1)d
Where a₁ is the first term and d is the common difference.
For the example above: a₅ = 3 + (5-1)×4 = 3 + 16 = 19 ✓
Sum of first n terms of an AP: Sₙ = n/2 × [2a₁ + (n-1)d]
Or equivalently: Sₙ = n/2 × (first term + last term) = n/2 × (a₁ + aₙ)
This elegant formula comes from clever pairing. If you add the first and last terms, second and second-to-last, etc., each pair sums to the same value. With n/2 such pairs, the total is n/2 times this value.
Example: Sum of 3, 7, 11, 15, 19: S₅ = 5/2 × (3 + 19) = 5/2 × 22 = 55
Check: 3 + 7 + 11 + 15 + 19 = 55 ✓
Geometric Progressions (GP)
A geometric progression is a sequence where consecutive terms have a constant common ratio r.
Example: 2, 6, 18, 54, 162, ... (r = 3)
General term: aₙ = a₁ × r^(n-1)
For the example: a₄ = 2 × 3^(4-1) = 2 × 27 = 54 ✓
Sum of first n terms of a GP:
- If r ≠ 1: Sₙ = a₁ × (1 - rⁿ) / (1 - r) = a₁ × (rⁿ - 1) / (r - 1)
- If r = 1: Sₙ = n × a₁ (trivial case: all terms equal a₁)
Example: Sum of 2, 6, 18, 54: S₄ = 2 × (3⁴ - 1) / (3 - 1) = 2 × 80 / 2 = 80
Check: 2 + 6 + 18 + 54 = 80 ✓
Infinite Geometric Series
When |r| < 1, an infinite geometric series converges (approaches a finite sum):
Sum of infinite GP: S∞ = a₁ / (1 - r) for |r| < 1
This is remarkable: infinitely many terms sum to a finite value.
Example: 1 + 1/2 + 1/4 + 1/8 + ... (r = 1/2) S∞ = 1 / (1 - 1/2) = 1 / (1/2) = 2
Visually, if you keep halving and adding, you approach 2 but never exceed it.
If |r| ≥ 1, the series diverges (doesn't approach a single value).
Harmonic Progressions
A harmonic progression (HP) is a sequence whose reciprocals form an arithmetic progression.
If {aₙ} is an HP, then {1/aₙ} is an AP.
Example: 1, 1/2, 1/3, 1/4, ... is harmonic because its reciprocals {1, 2, 3, 4, ...} form an AP.
There's no simple formula for the sum of HP terms, but understanding the reciprocal relationship is useful.
Real-World Applications
Compound Interest: With principal P, rate r, and n time periods: A = P(1 + r)ⁿ. This is a geometric progression.
Population Growth: Populations often grow geometrically. If growth rate is r%, the population follows a GP.
Loan Payments: Monthly mortgage payments form an arithmetic series. The total paid is the sum of all payments.
Salary Growth: Fixed annual raises create an AP; percentage raises create a GP.
Loan Amortization: Calculating remaining balance uses series summation.
Key Formulas
- AP general term: aₙ = a₁ + (n-1)d
- AP sum: Sₙ = n/2 × [2a₁ + (n-1)d]
- GP general term: aₙ = a₁ × r^(n-1)
- GP sum (finite): Sₙ = a₁(rⁿ - 1)/(r - 1)
- GP sum (infinite): S∞ = a₁/(1 - r) for |r| < 1
Socratic Questions
- In an arithmetic progression, why is the sum formula n/2 × (first + last) so elegant? What principle underlies this pairing method?
- A geometric progression with |r| < 1 sums to a finite value even with infinitely many terms. How is this possible? What happens at the boundary where |r| = 1?
- Compound interest involves geometric progressions. Why does "compounding" create exponential growth rather than linear growth? How does the time period affect the final amount?
- If you know the first term and the common difference of an AP, you can find any term directly. What information would you need about a GP to do the same? Why is finding specific terms different for these two types?
- Harmonic progressions are defined by their reciprocals forming an AP. Why define them this way? What advantages or insights does this reciprocal relationship provide?