What kind of sequence is the following?
In this question, we are given a graph with four points plotted on the 𝑥𝑦-coordinate plane. The first point has coordinates one, three, noting that each square on the 𝑦-axis represents two units. The second point has coordinates two, six. The third has coordinates three, 12. And the fourth point has coordinates four, 24. The 𝑥-coordinates here are the consecutive integers from one to four, and the corresponding 𝑦-values are three, six, 12, and 24.
We can consider this as the first four terms of a sequence. We know that an arithmetic sequence has a common difference between each of its terms. In this sequence, the difference between the first and second term is three. Between the second and third term, the difference is six. And between the third and fourth term, the difference is 12. This means that our sequence is not arithmetic. An arithmetic sequence would be represented by a linear or straight-line graph.
We recall that a geometric sequence has a common ratio, or multiplier, between consecutive terms. We know that three multiplied by two is equal to six. Six multiplied by two is equal to 12. And 12 multiplied by two is 24. This means that our sequence does have a common ratio. This is equal to two. As in this question, a geometric sequence will have an exponential graph. We can therefore conclude that the sequence shown in this question is geometric only.