Arithmetic and geometric progessions

Arithmetic progression (AP)

General term: nth term of an AP

aₙ = a + (n-1)d

Sum of first n terms of an AP:

      n
Sₙ = ─── [2a + (n-1)d]
      2

Geometric progression (GP)

Often abbreviated as GP.

 aᵢ₊₁
──────  =  r
  aᵢ

a, ar, ar², ar³, ...

General term: nth term of a GP

aₙ = arⁿ⁻¹

Sum of first n terms of a GP:

      a(1-rⁿ)
Sₙ = ───────── , could use if r < 1
       1 - r


      a(rⁿ-1)
Sₙ = ───────── , could use if r > 1
       r - 1


Sₙ = an        , if r = 1

provided no term is zero.

Sum of an infinite GP ²:

       a      
S = ─────── , where |r| < 1
     1 - r

Arithmetic and geometric means

Arithmetic mean of two numbers a and b:

      a + b    
AM = ───────
        2

Geometric mean of two positive numbers a and b:

GM = √(ab)

A relation: AM ≥ GM

References