To determine which recursive equation correctly describes Adrian's annual income given that his salary is 1.05 times the previous year's salary and his initial salary is [tex]$52,000, let's break down the components of the problem.
1. Initial Salary:
Adrian's salary at the start (year 1) is $[/tex]52,000. Therefore:
[tex]\[
f(1) = 52,000
\][/tex]
2. Growth Multiplier:
Each year, Adrian's salary is 1.05 times the salary from the previous year. Therefore:
[tex]\[
f(n) = 1.05 \cdot f(n-1) \quad \text{for } n \geq 2
\][/tex]
Now, let's analyze the given options to see which one fits these components:
- Option A:
[tex]\[
f(1) = 52,000
\][/tex]
[tex]\[
f(n) = 1.05 + f(n-1) \quad \text{for } n \geq 2
\][/tex]
This option incorrectly states that the salary increases by adding 1.05 instead of multiplying by 1.05. This is not consistent with the problem statement.
- Option B:
[tex]\[
f(1) = 1.05
\][/tex]
[tex]\[
f(n) = 52,000 \cdot f(n-1) \quad \text{for } n \geq 2
\][/tex]
This option incorrectly sets the initial salary to 1.05 instead of 52,000. Furthermore, it incorrectly multiplies 52,000 by the function value from the previous year.
- Option C:
[tex]\[
f(1) = 1.05
\][/tex]
[tex]\[
f(n) = 52,000 + f(n-1) \quad \text{for } n \geq 2
\][/tex]
This option incorrectly sets the initial salary to 1.05 instead of 52,000. Additionally, it incorrectly adds 52,000 to the function value from the previous year.
- Option D:
[tex]\[
f(1) = 52,000
\][/tex]
[tex]\[
f(n) = 1.05 \cdot f(n-1) \quad \text{for } n \geq 2
\][/tex]
This option correctly sets the initial salary to 52,000 and states that each subsequent year's salary is 1.05 times the previous year's salary.
Given the requirements and the correct initial salary and growth formula, Option D is the correct answer. Therefore:
[tex]\[
\boxed{D}
\][/tex]
That is:
[tex]\[
f(1) = 52,000
\][/tex]
[tex]\[
f(n) = 1.05 \cdot f(n-1) \quad \text{for } n \geq 2
\][/tex]
