To determine which recursive formula can be used to represent the total amount of time Zina spends making hats based on the total amount of time spent previously, let's analyze the sequence provided:
[tex]$
1.5, 2.25, 3.0, 3.75, \ldots
$[/tex]
1. Identify the Pattern:
Let's look at the differences between consecutive terms in the sequence:
- [tex]\( 2.25 - 1.5 = 0.75 \)[/tex]
- [tex]\( 3.0 - 2.25 = 0.75 \)[/tex]
- [tex]\( 3.75 - 3.0 = 0.75 \)[/tex]
We observe that the sequence is increasing by a constant value of 0.75.
2. Understand the Recursive Formula:
- A recursive formula expresses the next term of a sequence in terms of the previous term.
- For an arithmetic sequence (where the difference between consecutive terms is constant), the recursive formula is of the form:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\( d \)[/tex] is the common difference.
3. Apply our Observation:
Given that the common difference [tex]\( d \)[/tex] in our sequence is 0.75, the recursive formula will have the form:
[tex]\[
f(n+1) = f(n) + 0.75
\][/tex]
Among the given options, the correct formula is:
[tex]\[
f(n+1) = f(n) + 0.75
\][/tex]
Therefore, the recursive formula that can be used to determine the total amount of time spent making hats based on the total amount of time spent previously is:
[tex]\[
f(n+1) = f(n) + 0.75
\][/tex]
