Geometric Progression (GP)

Geometric Progression (GP)

This lesson covers the following key concepts:

• Definition and general term of GP
• Common ratio and its properties
• nth term formula: aₙ = arⁿ⁻¹
• Sum of n terms of finite GP
• Sum to infinity (|r| < 1)
• Properties of GP (three terms in GP, geometric mean)
• Insertion of geometric means between two numbers

Important Formulas

• $$a_n = ar^{n-1}$$
• $$S_n = \frac{a(1-r^n)}{1-r} = \frac{a(r^n - 1)}{r - 1}$$
• $$S_\infty = \frac{a}{1-r} \text{ for } |r| < 1$$
• $$\text{GM between } a, b: \sqrt{ab}$$
• $$\text{If } a, b, c \text{ in GP: } b^2 = ac$$
• $$\text{Insert m GMs: } r = \left(\frac{b}{a}\right)^{1/(m+1)}$$