Geometric progressions

A geometric progression is such a sequence of numbers that the quotient of any two successive members of the sequence is a constant, which is called ratio of the progression.

Example: 3, 6, 12, 24 …. r = 2

General term: a_n = a₁· r^n-1

In the example: a_n = 3 · 2^n-1

Other examples:

•40, 20, 10, 5, …. a_n = 40 · 0,5^n-1

•8, -16, 32, -64, … a_n = 8 · (-2)^n-1

Exercises:

1.- Find the general term and a₆ of these progressions:

a) 3, 6, 12, 24,...

b) 2, -4, 8, -16,...

c) 9, 3, 1, 1/3,...

d) a₁= 5, a₃ = 45

2.- A kind of bacterium reproduces by bipartition every 15 minutes. How many bacteria will there be after 6 hours?

Solutions: 1.-a) a_n = 3·2^n-1; a₆ = 96; b) a_n = 2·(-2)^n-1 = (-1)^n-1·2ⁿ, a₆ = -64; c) a_n = 9·(1/3)^n-1 = 3^3-n, a₆ = 3^-3 = 1/27;

d) Two posibilities: a_n = 5·3^n-1, a₆ = 1215, b_n = 5·(-3)^n-1, b₆ = -1215

2.- 2²³ bacteria = 8388608 bacteria