Definition of a Sequence
A sequence is a list of numbers, or more formally, a function
f(n) from the
natural numbers to the real numbers. We write
a1 = 1/3, a2 = 1/4, etc.
Write the general term an for the following sequences:
The Limit of a Sequence
Consider the sequence
1/2, 2/3, 3/4, 4/5, ...
We see that as n becomes large the numbers approach 1.
If there is no such L then we say that the sequence diverges.
Find the limit of the following sequences or state that the sequences diverge.
Currently there are 5,000 of a certain species of monkeys left in a jungle. Each year, because of poachers and loss of habitat, the population decreases by 5%. Write a sequence that models this and find the limit of the sequence.
a0 = 5000 a2 = 5000 - 5000(.05) = 5000(.95)
a2 = 5000(.95) - 5000(.95)(.05) = 5000(.95)(.95) = 5000(.95)2
Following this pattern, we get
an = 5000(.95)n
We find the limit of this sequence to be 0, that is the population will eventually become extinct.