Question

The first three terms of a sequence are plotted on the coordinate plan show below, where x represents the actual terms of the sequence.

Answer the following four questions to prove that the sequence is an arithmetic sequence.

Questions:
1. Is there a constant rate of change between consecutive terms of the sequence? If so, what is it?

2. Give the ordered pair that would represent the next term in this sequence.

3. If this ordered pair were graphed. what pattern would all four points follow?

4. Can this sequence be represented with an explicit function? If so, write the function.​

Answer 1

Is there a constant rate of change between consecutive terms of the sequence? If so, what is it?

To check for a constant rate of change, we can calculate the difference between consecutive y-values (terms):

From (2, 1) to (4, 2): Δy=2−1=1

From (4, 2) to (6, 3): Δy=3−2=1

The difference is consistently 1, indicating a constant rate of change.

Give the ordered pair that would represent the next term in this sequence.

If the rate of change is constant, the next term would be obtained by adding 1 to the y-value of the last given term. So, the next term is (8, 4).

If this ordered pair were graphed, what pattern would all four points follow?

The pattern of the points forms a straight line, suggesting that this sequence is an arithmetic sequence.

Can this sequence be represented with an explicit function? If so, write the function.

Yes, since the rate of change is constant, the sequence can be represented by an arithmetic function. The general form of an arithmetic sequence is y=mx+b, where m is the slope (rate of change) and b is the y-intercept. In this case, the function is y=x+1.