### Video Transcript

For the next three years, the enrollment at a university is expected to increase by 55 students each year. If the university’s current enrollment is 589 students, determine their enrollment in each of the next three years.

In this question, we’re given some information about the enrollment at a university. We’re told for the next three years, the enrollment is going to increase by 55 students every year. And what this tells us is for the next three years, every year there’s going to be 55 more students enrolled than the last year. We’re also told that the current enrollment is 589 students. We need to use this information to determine the enrollment number in each of the next three years. To answer this question, let’s start by looking at all the information we’re given.

First, we’re told that currently there are 589 students enrolled at the university. Next, we’re also told that for the next three years, the number of students enrolled is going to increase by 55 each year. And there’s two different ways we could think about this number. We could think about this is an increase of 55 each year. So to answer this question, we just need to add 55 to our value three times. However, we can also think about this as the difference between enrollment rates in consecutive years.

We’re told that for the next three years, this is a constant value of 55. And this tells us something useful about the number of students involved in this university year by year. Since for these four years, the difference between consecutive terms is constant. We call this an arithmetic sequence, where the initial value or zeroth term of this sequence is 589 and the common difference, 𝑑, is going to be equal to 55 because we’re increasing by 55 each year.

And this gives us another way of answering this question because we know how to generate the 𝑛th term of an arithmetic sequence. For an arithmetic sequence with common difference 𝑡 and initial value 𝑎, the 𝑛th term is given by 𝑑 times 𝑛 plus 𝑎. We can use this to find the number of students enrolled at our university for the next three years. For our arithmetic sequence, the value of 𝑎 is 589 and the common difference 𝑑 is equal to 55.

So, we can calculate the number of students enrolled after one year by substituting 𝑛 is equal to one into this expression. we get 55 times one plus 589. And if we calculate this, we get 644. We can do the same to find the number of students enrolled after two years. We use 𝑛 is equal to two. We get 55 times two plus 589, which we can calculate is 699.

Finally, we’ll do exactly the same for the number of students enrolled after three years. We substitute 𝑛, 𝑛 is equal to three. This gives us 55 times three plus 589, which we can calculate is equal to 754. Therefore, we found the enrollment of this university for the next three years. We get in the first year 644, in the second year 699, and in the third year 754.

However, there is one thing worth pointing out about our formula for finding the number of students enrolled after 𝑛 years. In this case, our formula is only valid up to 𝑛 is equal to three. This is because in the question, we’re only told that the number of students is going to increase by 55 for the next three years. We don’t know what will happen after four years or five years or any further years.

So, sometimes, in questions like this, we do need to be careful about the exact information we’re given. For example, we would not be able to use this formula to find the number of students enrolled at this university after 10 years. However, we were able to show that in the next three years, the enrolment at this university will be equal to 644 after one year, 699 after two years, and 754 after three years.