Arithmetic

Geometric Progression

The Chessboard Legend: The Inventor Who Outsmarted the King

The Indian sage **Sissa** (by legend) invented chess for King Sheram. The delighted ruler offered any reward. Sissa asked modestly: **one grain** on the first square of the board, two on the second, four on the third - doubling all the way to the 64th square.

Mighty lord, I want neither gold nor palaces. Give me only grains of wheat. - Sissa ibn Dahir

This story is the earliest known description of exponential growth. Biologists use it to explain pandemics, financiers use it for compound interest, programmers use it to understand O(2ⁿ) complexity. The human mind is bad at grasping exponentiation. But the chessboard legend is never forgotten.

Place 1 grain on the first square of a chessboard, 2 on the second, 4 on the third... How many on all 64 squares? More than the world's wheat harvest for 1000 years! Geometric progression is exponential growth - the most powerful force in mathematics.

  • **Finance:** compound interest, discounting
  • **Biology:** population growth, virus spread
  • **Physics:** radioactive decay, damping

What is a Geometric Progression?

A **geometric progression** (GP) is a sequence where the ratio of consecutive terms is constant. Each term is obtained by multiplying the previous one by q.

**Definition:** a, aq, aq², aq³, ... **Parameters:** • a₁ = a - first term • q - common ratio **GP condition:** aₙ₊₁ / aₙ = q (constant)

GPs model exponential processes: population growth, compound interest, radioactive decay.

What is the common ratio q in the progression 5, 15, 45, 135, ...?

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Formula for the nth Term

The **nth term formula** for a GP involves a power of q, giving exponential growth (or decay).

**Formula:** aₙ = a₁ × q^(n-1) **Logarithmic form:** log(aₙ) = log(a₁) + (n-1)log(q) In logarithmic scale, a GP is linear!

Exponential growth - slow at first, then explosive. This explains pandemics, viral spread, and compound interest.

What is the 8th term of the progression 1, 3, 9, 27, ...?

Sum of a Geometric Progression

The **GP sum** has a compact formula derived using an elegant multiplication trick.

**Sum formula (q ≠ 1):** Sₙ = a₁ × (qⁿ - 1) / (q - 1) **Or:** Sₙ = a₁ × (1 - qⁿ) / (1 - q) **If q = 1:** Sₙ = n × a₁

The sum of a GP grows nearly as fast as its last term (when q > 1). That's the essence of exponential growth.

What is the sum 1 + 3 + 9 + 27 + 81?

Infinite Geometric Progression

If |q| < 1, the sum of an **infinite GP** is finite! This is one of the first examples of a convergent series.

**Sum of infinite GP (|q| < 1):** S∞ = a₁ / (1 - q) **Convergence condition:** |q| < 1 (qⁿ → 0 as n → ∞)

The infinite GP is the bridge between discrete and continuous. It shows that the infinite can be finite.

The sum of infinitely many terms is always infinite

If the terms decrease fast enough (|q| < 1), the sum is finite

1/2 + 1/4 + 1/8 + ... = 1. Each term is half the previous. The total never exceeds 2 × the first term. This is counterintuitive: infinitely many small numbers can add up to a finite value.

What is the sum 1 + 1/3 + 1/9 + 1/27 + ...?

Key Ideas

  • GP: aₙ₊₁/aₙ = q (constant)
  • nth term: aₙ = a₁ × q^(n-1)
  • Sum: Sₙ = a₁(qⁿ-1)/(q-1)
  • Infinite GP when |q|<1: S∞ = a₁/(1-q)

Related Topics

GP connects to powers and exponentials:

  • Powers — nth term is a power of q
  • Arithmetic Progression — Addition instead of multiplication
  • Fibonacci Numbers — Ratio → φ (golden ratio)

Вопросы для размышления

  • Why does exponential growth seem slow at first?
  • How are geometric progressions related to compound interest?
  • Why does 0.999... = 1, and not "slightly less"?

Связанные уроки

  • calc-01-sequences
  • calc-02-series-intro
Geometric Progression