a. Write a recursive formula for the number of wolves
b. Write an explicit formula for the number of wolves
c. If this trend continues, how many wolves will there be in 10 years?
Answer:
a). [tex]P_{n}=(1.35)P_{n-1}[/tex]
b). [tex]P_{n}=200(1.35)^n[/tex]
c). 4021 wolves
Step-by-step explanation:
Originally number wolves transplanted in the forest = 200
After 3 years, population of the wolves grown = 270
As we know population growth is an exponential phenomenon.
Therefore, sequence formed will be a geometric sequence.
If [tex]P_{0}[/tex] is the first term and [tex]P_{1}[/tex] is the successive term, then [tex]P_{1}=P_{0}(1+r)[/tex]
where r is the common ratio of each term.
[tex]270=200(1+r)[/tex]
(1 + r) = [tex]\frac{270}{200}[/tex]
r = 1.35 - 1
r = 0.35
a). Recursive formula for the number of wolves will be
[tex]P_{n}=P_{n-1}(1+r)[/tex]
[tex]P_{n}=P_{n-1}(1+0.35)[/tex]
[tex]P_{n}=(1.35)P_{n-1}[/tex]
b). Explicit formula of a exponential sequence is given by
[tex]P_{n}=P_{0}(1+r)^n[/tex]
[tex]P_{n}=200(1.35)^n[/tex]
c). We have to calculate the number of wolves in 10 years.
[tex]P_{n}=200(1.35)^{10}[/tex]
= 200×(20.1066)
= 4021 wolves
