A simple but interesting result of Galileo

 

Nature's great book is written in mathematics.  -Galileo

 

[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those
properties which we give to the finite and limited; -Galileo 

 

 

Sequences have always been of great interest to mathematicians. 

 

There are the simplest arithmetic sequences where the difference between the terms is a constant such as 2,4,6,8,10,12, ...

There are sequences called geometric sequences where consecutive terms have a constant ratio like 1,2,4,8,16,32,.... 

There are sequences that follow a rule like each term is the sum of the previous two terms (the Fibonacci sequence) 1,1,2,3,5,8,...

and the list goes on and on.

 

Below we examine a rather simple sequence 1/3, 1/3, 1/3, ... what is interesting is how each term is formed.

In 1615, while studying falling objects Galileo made a surprising (at least to me) mathematical discovery.  He found that

 

 

that is

 

   

 

  1. Prove Galileo’s discovery.
  2. First create another (obvious) sequence that is created similarly to the one of Galileo.
  3. Does this new sequence have the same properties i.e. does b1= b2= b3= ...?
  4. In the sequence of Galileo, what happens as n approaches infinity?
  5. What happens as n approaches infinity for your new sequence?
  6. Can you generalize this result?
  7. Make at least 4 similar conjectures (the bigger the better).