Answer:
Here's what I find.
Step-by-step explanation:
You have 800 deer at the end of Year 1, and you expect the population to decrease each year thereafter.
a) i) The recursive formula
Let dâ‚™ = the deer population n years after the initial measurement.
[tex]d_{n} = d_{n - 1}r^{n}[/tex]
For this situation,
[tex]d_{n} = d_{n - 1}(0.5)^{n}[/tex]
a) ii) Definitions
n = the number of years from first measurement
r = the common ratio, that is, the deer population at the end of one year divided by the population of the previous year.
a) iii) First term of sequence
The first term of the sequence is dâ‚€, the population when first measured.
b) The function formula
The formula for the nth term of a geometric series is
[tex]d_{n} =d_{0}r^{n - 1}[/tex]
c) Value of dâ‚€
Let n = 2; then dâ‚‚ = 800
[tex]\begin{array}{rcl}800 & = & d_{0}(0.5)^{2 - 1}\\800 & = & d_{0}(0.5)\\\\d_{0} & = & \dfrac{800}{0.5}\\\\& =&\mathbf{1600}\\\end{array}[/tex]
The recursive sequence is [tex]a_n = 0.5a_{n -1}[/tex]
The given properties are:
The initial population (a) = 800
The rate of decline (r) = 50%
Given that the number of deer reduces, the recursive function that defines the nth term of the sequence is:
[tex]a_n = a_{n -1} * (1 - 50\%)\\ \\[/tex]
[tex]a_n = 0.5a_{n -1}[/tex]
Hence, the recursive sequence is [tex]a_n = 0.5a_{n -1}[/tex]
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