the sequence below represents marisas fine at the library for each day that she has an overdue book: $0.50…

Explanation:

Step1: Identify the sequence type

The sequence of fines is $0.50, 0.65, 0.80, 0.95, 1.10, \dots$. This is an arithmetic sequence where the first term $a_1 = 0.50$ and the common difference $d$ is calculated as $0.65 - 0.50 = 0.15$, $0.80 - 0.65 = 0.15$, etc. So the common difference $d = 0.15$.

Step2: Recall the arithmetic sequence formula

The formula for the $n$th term of an arithmetic sequence is $a_n = a_1 + (n - 1)d$. Here, the function $f(n)$ represents the fine on the $n$th day, so we substitute $a_1 = 0.50$ and $d = 0.15$ into the formula: [ \begin{align*} f(n)&= 0.50 + (n - 1) \times 0.15\ &= 0.50 + 0.15n - 0.15\ &= 0.15n + 0.35 \end{align*} ]

We can also check by plugging in $n = 1$: $f(1)=0.15(1)+0.35 = 0.50$, which matches the first term. For $n = 2$: $f(2)=0.15(2)+0.35 = 0.30 + 0.35 = 0.65$, which matches the second term.

Answer:

$f(n) = 0.15n + 0.35$ (the third option: $f(n)=0.15n + 0.35$)