Question

a 10th - grade class at the local high school spent the day at an amusement park. in the first hour, they rode 3 rides. after 2 hours, they rode 5 rides. they had ridden 7 rides after 3 hours.

• create an arithmetic sequence for the nth term of the sequence

• find the 9th term of the sequence and interpret the solution in terms of the situation

show your work or explain how you found your answer. enter your answers and your work or explanation in the space provided

Explanation:

Step1: Identify first - term and common - difference

The first - term $a_1$ of the arithmetic sequence is the number of rides in the first hour. So $a_1 = 3$. The number of rides increases by 2 each hour (since from 1 hour to 2 hours, it goes from 3 to 5 rides, a difference of 2; from 2 hours to 3 hours, it goes from 5 to 7 rides, a difference of 2). So the common - difference $d=2$.

Step2: Write the formula for the nth term of an arithmetic sequence

The formula for the nth term of an arithmetic sequence is $a_n=a_1+(n - 1)d$. Substituting $a_1 = 3$ and $d = 2$ into the formula, we get $a_n=3+(n - 1)\times2$. Simplify this formula:
\[

$$\begin{align*} a_n&=3 + 2n-2\\ a_n&=2n + 1 \end{align*}$$

\]

Step3: Find the 9th term of the sequence

Substitute $n = 9$ into the formula $a_n=2n + 1$.
\[

$$\begin{align*} a_9&=2\times9+1\\ a_9&=18 + 1\\ a_9&=19 \end{align*}$$

\]

Step4: Interpret the 9th term

In the context of the situation, the 9th term represents the number of rides the class will have ridden after 9 hours at the amusement park. So after 9 hours, the class will have ridden 19 rides.

Answer:

The formula for the nth term of the arithmetic sequence is $a_n=2n + 1$. The 9th term of the sequence is 19, which means that after 9 hours at the amusement park, the class will have ridden 19 rides.