Question

1) The deer population in a national park is expected to decline over time. The park

Answer 1

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 dₙ = the deer population n years after the initial measurement.

$d_{n} = d_{n - 1}r^{n}$

For this situation,

$d_{n} = d_{n - 1}(0.5)^{n}$

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 d₀, the population when first measured.

b) The function formula

The formula for the nth term of a geometric series is

$d_{n} =d_{0}r^{n - 1}$

c) Value of d₀

Let n = 2; then d₂ = 800

$\begin{array}{rcl}800 & = & d_{0}(0.5)^{2 - 1}\\800 & = & d_{0}(0.5)\\\\d_{0} & = & \dfrac{800}{0.5}\\\\& =&\mathbf{1600}\\\end{array}$