Recursively-Defined Sequences
In mathematical models for science, it is often the case that a sequence is defined not in terms
of the term number, but rather in terms of the current value. If you want to predict the amount of
bacteria in a culture tomorrow, you need to know how much is in the culture today, but you don’t
need to know today’s date. For example, suppose a colony of bacteria doubles each day. Then the
mathematical statement that describes this fact is
Nt+1 = 2Nt.
We know from an earlier worksheet that Nt+1/Ntconstant and greater than 1 means that the
sequence is growing exponentially, although we have not used an exponential function to define
the sequence. A sequence defined in terms of earlier values is said to be defined recursively.
Recursively-defined sequences can have interesting properties.
1. Define a sequence by xt+1 = 61−xtxtalong with the starting value x0= 0.8. Then
x1= 61−x0x0= 0.8(60.2)≈1.145.
Determine the terms from t= 2 to t= 8.
What do you think happens to this sequence as n→ ∞? Explain your answer.
2. Define a sequence by xt+1 = 91−xtxtalong with the starting value x0= 0.8. Determine the
terms from t= 1 to t= 8.
What do you think happens to this sequence as n→ ∞? Explain your answer.