Geometric Progression

Definition

A geometric progression (GP) is a sequence in which, starting from the second term, each term is obtained by multiplying the previous one by the same number q (b₁ ≠ 0, q ≠ 0). q is the common ratio: q = bₙ₊₁ / bₙ.

• Example: 3, 6, 12, 24, ... where b₁ = 3, q = 2.
• If |q| > 1, the terms grow in absolute value; if |q| < 1, they decrease; if q < 0, the signs alternate.

Main Formulas

What is found	Formula
n-th term	bₙ = b₁ · qⁿ⁻¹
Sum of the first n terms (q ≠ 1)	Sₙ = b₁ · (qⁿ − 1) / (q − 1)
If q = 1	Sₙ = n · b₁
Characteristic property	bₙ² = bₙ₋₁ · bₙ₊₁
Infinite decreasing GP (−1 < q < 1)	S = b₁ / (1 − q)

• If three consecutive positive numbers a, b, c form a GP, then b² = ac.
• bₙ = bₘ · qⁿ⁻ᵐ. If two terms are given, qⁿ⁻ᵐ = bₙ / bₘ.
• If the sums of the indices are equal, the products of the terms are equal: if k + m = p + r, then bₖ · bₘ = bₚ · bᵣ.

Solution Procedure

1. Find b₁ and q (if needed, from bₙ / bₘ)
2. Choose the required formula: bₙ, Sₙ or the infinite sum
3. Substitute the values and calculate
4. Check the result against the condition

Examples

• If b₁ = 2, q = 3, then b₄ = 2 · 27 = 54 and S₄ = 2 · (81 − 1) / 2 = 80.
• b₂ = 6, b₅ = 162: q³ = 162 / 6 = 27, so q = 3 and b₁ = 2.
• In 6, 2, 2/3, ... we have q = 1/3, S = 6 / (1 − 1/3) = 9.

Mistakes

• The infinite sum exists only for −1 < q < 1.
• The exponent is n − 1, not n: b₅ = b₁ · q⁴.
• When finding the ratio, q = next term / previous term.