Exercise 45 Page 626

We are asked to determine if the geometric mean and the mean are equal for two consecutive positive integers. Let's recall the formulas for the geometric mean and the mean of two positive integers a and b.

Let x and x+1 be two consecutive positive integers. Our next step will be to evaluate both means for these numbers and check if they are equal. We will start with the geometric mean.

Next, let's evaluate the arithmetic mean.

Let's set these two expressions together and check if they are equal.

Since we ended with a false statement, we can say that the geometric mean for consecutive integers is never the mean of the two numbers.