Exercise 2 Page 204

Essentially, we are given four different questions and asked to find the solutions to each one. One of the four questions will provide a different answer than the other three. Let's look at each question individually and then compare.

First question

Find the slope of the linear function.

To find the slope of the function, we need to choose two arbitrary points and use the Slope Formula. Let's use the first two, $(2,10)$ and $(3,13).$ The slope of the linear function is $3.$

Second question

Find the difference between consecutive terms of the arithmetic sequence.

The $x$ values of the coordinate pairs represent $n,$ the number of the term within the sequence, $n=1$ is the first term, $n=2$ is the second term, and so on. Because the values for $n$ are $2,$ $3,$ $4,$ and $5,$ these are all consecutive terms. We can find the difference by subtracting the $y$ values of the ordered pairs. The difference between each pair of consecutive terms is $3.$

Third question

Find the difference between the terms $a_2$ and $a_4.$

To find the difference between $a_2$ and $a_4,$ first we need to find the value of these terms. $a_2$ indicates the $y\text{-}$coordinate for the second term, which is $10.$ Similarly, $a_4$ indicates the $y\text{-}$coordinate for the fourth term, which is $16.$ The difference is then:

Fourth question

Find the common difference of the arithmetic sequence.

The common difference is the number of units that separates each pair of consecutive terms in a arithmetic sequence. Because the values for $n$ are $2,$ $3,$ $4,$ and $5,$ these are all consecutive terms. We can check the difference between each pair. The difference is always $3.$ Therefore, the common difference $d$ is $3.$

Conclusion

The only question with a different answer was the third, the difference between the second and fourth terms. The slope of the linear function representing an arithmetic sequence is the same thing as the difference between consecutive terms which is just another way to describe the common difference.