Let n be a positive integer. How many integers are there between the square root of n squared plus n plus 1 and the square root of 9n squared plus n plus 1? So first off, square roots are difficult, but square numbers are a lot easier. So instead of counting the number of integers between these two square root expressions, we can just square everything and count the number of square numbers between these expressions without the square root. So n squared plus n plus 1 is in between n squared and n plus 1 squared, because this guy is equal to n squared plus 2n plus 1. And now 9n squared plus n plus 1 is greater than 9n squared, which is 3n squared, and it's also less than 3n plus 1 squared, because this guy is equal to 9n squared plus 6n plus 1. So we want to count all the square numbers starting from n plus 1 squared all the way up to 3n squared. Since n is a positive integer, 3n is always bigger than n plus 1. And so the total number of integers is 3n minus n plus 1 plus 1, because we don't want to miss the endpoints, which is 2n. And so our final answer is here, 2n. Now, on the exam, there is another way to solve this problem. The first step is to notice that all of these answers are linear functions of n. And in order to determine a line, you need exactly two points, meaning you can just do the calculation when n equals 1 and n equals 2, and that will tell you exactly which line you're on. So that's just another way to solve it during an exam.

square root expressions

perfect squares

integer count

linear relationship

mathematical problem