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Zina spends 1.5 hours setting up her sewing machine and making one hat. The total amount of time spent making hats can be represented by the sequence below.
[tex]\[ 1.5, 2.25, 3.0, 3.75, \ldots \][/tex]

Which recursive formula can be used to determine the total amount of time spent making hats based on the total amount of time spent previously?

A. [tex]\( f(n+1) = f(n) + 1.5 \)[/tex]
B. [tex]\( f(n+1) = f(n) + 0.75 \)[/tex]
C. [tex]\( f(n+1) = \frac{1}{2} f(n) \)[/tex]
D. [tex]\( f(n+1) = \frac{3}{2} f(n) \)[/tex]



Answer :

GinnyAnswer
To determine the recursive formula representing the total amount of time Zina spends making hats, we need to first observe the given sequence of times:

[tex]\[ 1.5, 2.25, 3.0, 3.75, \ldots \][/tex]

Next, let's examine how each term in the sequence relates to the previous term. Specifically, we need to identify the difference between consecutive terms to determine the pattern.

1. The first term is [tex]\( 1.5 \)[/tex].
2. The second term is [tex]\( 2.25 \)[/tex]. The difference between the first and second term is:
[tex]\[ 2.25 - 1.5 = 0.75 \][/tex]
3. The third term is [tex]\( 3.0 \)[/tex]. The difference between the second and third term is:
[tex]\[ 3.0 - 2.25 = 0.75 \][/tex]
4. The fourth term is [tex]\( 3.75 \)[/tex]. The difference between the third and fourth term is:
[tex]\[ 3.75 - 3.0 = 0.75 \][/tex]

From the sequence, we see that each term increases by a constant value of [tex]\( 0.75 \)[/tex].

A sequence that increases by a constant value can be described by a linear recursive formula. Thus, the recursive formula for this sequence adds [tex]\( 0.75 \)[/tex] to the previous term to get the next term.

The recursive formula is:
[tex]\[ f(n+1) = f(n) + 0.75 \][/tex]

Therefore, among the given options, the correct recursive formula is:
[tex]\[ f(n+1) = f(n) + 0.75 \][/tex]

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