To determine the recursive formula for the electrician’s earnings based on the number of hours worked, let’s analyze the pattern in the given sequence:
The earnings sequence is:
[tex]\[ 110, 130, 150, 170, \ldots \][/tex]
1. Identifying the pattern:
- For the first hour, the electrician earns [tex]$110.
- For the second hour, the electrician earns $[/tex]130.
- For the third hour, the electrician earns [tex]$150.
- For the fourth hour, the electrician earns $[/tex]170.
2. Finding the difference between successive terms:
[tex]\[ 130 - 110 = 20 \][/tex]
[tex]\[ 150 - 130 = 20 \][/tex]
[tex]\[ 170 - 150 = 20 \][/tex]
We see that the difference between the earnings for each successive hour is [tex]$20. This indicates that each hour the electrician earns $[/tex]20 more than the previous hour.
3. Defining the recursive formula:
Let [tex]\( f(n) \)[/tex] represent the total amount of money earned after [tex]\( n \)[/tex] hours. According to our observations, the amount earned after [tex]\( n+1 \)[/tex] hours can be obtained by adding $20 to the amount earned after [tex]\( n \)[/tex] hours. Therefore, the recursive formula should be:
[tex]\[ f(n+1) = f(n) + 20 \][/tex]
Hence, the correct recursive formula for the amount of money earned for each successive hour worked is:
[tex]\[ \boxed{f(n+1) = f(n) + 20} \][/tex]
