Question

1. professor jones is collecting data on the number of students who arrive late to his history of yoyos lectures. the table shows the data he has collected for his first 5 lectures.

a. is this sequence arithmetic or geometric? (circle one)
b. write an equation that models the number of students at each lecture.
c. how many students will be late to his 10th lecture?
d. if there are 100 students in his class, how many lectures until every student is late, assuming the same rate continues?

Explanation:

Step1: Check for arithmetic - common difference

Find the differences between consecutive terms of the sequence of late - students.
$3 - 0=3$, $6 - 3 = 3$, $9 - 6=3$, $12 - 9 = 3$. Since the common difference $d = 3$, the sequence is arithmetic.

Step2: Find the equation of the arithmetic sequence

The general form of an arithmetic sequence is $y=a+(x - 1)d$, where $a$ is the first - term and $d$ is the common difference. Here, $a = 0$ and $d = 3$. So the equation is $y=0+(x - 1)\times3=3x-3$.

Step3: Find the number of late students for the 10th lecture

Substitute $x = 10$ into the equation $y = 3x-3$.
$y=3\times10-3=27$.

Step4: Find the number of lectures until all 100 students are late

Set $y = 100$ in the equation $y = 3x-3$ and solve for $x$.
$100=3x - 3$.
Add 3 to both sides: $100 + 3=3x$, so $103=3x$.
Then $x=\frac{103}{3}\approx34.33$. Since the number of lectures must be a whole number, and we need all 100 students to be late, we take the next whole number, so $x = 35$.

Answer:

a. Arithmetic
b. $y = 3x-3$
c. 27
d. 35