# 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} = a_{1}· 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 a_{6} of these progressions:

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

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

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

d) a_{1}= 5, a_{3} = 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}; a_{6} = 96; b) a_{n} = 2·(-2)^{n-1} = (-1)^{n-1}·2^{n}, a_{6} = -64; c) a_{n} = 9·(1/3)^{n-1} = 3^{3-n}, a_{6} = 3^{-3} = 1/27;

d) Two posibilities: a_{n} = 5·3^{n-1}, a_{6} = 1215, b_{n} = 5·(-3)^{n-1}, b_{6} = -1215

2.- 2^{23} bacteria = 8388608 bacteria

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