To solve this problem, we need to describe the population decrease using a recursively-defined function. We are given that the initial population after 1 year is 14,000 birds and the population decreases by 10% each year.
1. First, we need to define the initial condition of the recursive function, which states the population at year 1:
[tex]\[
f(1) = 14000
\][/tex]
2. Next, we need to describe the recurrence relation that captures the population decrease each year. Since the population decreases by 10% annually, which is the same as multiplying by [tex]\( 0.90 \)[/tex] each year:
[tex]\[
f(n) = f(n-1) \times (1 - 0.10), \text{ for } n \geq 2
\][/tex]
Putting it all together we get:
[tex]\[
f(1) = 14000
\][/tex]
[tex]\[
f(n) = f(n-1) \times 0.90, \text{ for } n \geq 2
\][/tex]
3. Finally, after 4 years, we use the recurrence relation to find the remaining bird population:
[tex]\[
f(4) = 10206 \text{ birds}
\][/tex]
Therefore, the correct answer in each box is:
- [tex]\( f(1) = 14000 \)[/tex]
- [tex]\( f(n) = f(n-1) \times 0.90 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- After 4 years, [tex]\( 10206 \)[/tex] birds will remain.
