To determine which recursive equation best represents Adrian's annual income given that his initial salary is [tex]$52,000 and it increases by a factor of 1.05 each year, let's break down the required elements of a correct recursive formula:
1. The initial year's salary \( f(1) \) should be directly specified as \$[/tex]52,000.
2. Each following year's salary [tex]\( f(n) \)[/tex] should be calculated by multiplying the previous year's salary [tex]\( f(n-1) \)[/tex] by the growth factor 1.05.
Given these conditions, let's examine each option:
Option A:
[tex]\[ f(1) = 1.05 \][/tex]
[tex]\[ f(n) = 52000 \cdot f(n-1) \,\, \text{for} \,\, n \geq 2 \][/tex]
This option does not correctly specify the initial salary, and the formula for future salaries improperly multiplies by \[tex]$52,000 rather than using the previous year's salary. This option is incorrect.
Option B:
\[ f(1) = 52,000 \]
\[ f(n) = 1.05 + f(n-1) \,\, \text{for} \,\, n \geq 2 \]
This option correctly specifies the initial salary, but it incorrectly adds the growth factor of 1.05 to the previous year's salary instead of multiplying. This option is incorrect.
Option C:
\[ f(1) = 1.05 \]
\[ f(n) = 52,000 + f(n-1) \,\, \text{for} \,\, n \geq 2 \]
This option does not correctly specify the initial salary, and it improperly adds \$[/tex]52,000 to the previous year's salary. This option is incorrect.
Option D:
[tex]\[ f(1) = 52,000 \][/tex]
[tex]\[ f(n) = 1.05 \cdot f(n-1) \,\, \text{for} \,\, n \geq 2 \][/tex]
This option correctly specifies the initial salary as \$52,000 and then correctly defines the subsequent year's salary as 1.05 times the previous year's salary.
Thus, Option D is the correct answer:
[tex]\[ f(1) = 52,000 \][/tex]
[tex]\[ f(n) = 1.05 \cdot f(n-1) \,\, \text{for} \,\, n \geq 2 \][/tex]
