a Let's analyze each graph, starting with

1 .

Graph 1

Examining the graph, it looks like it might be arithmetic. If it is, the rate-of-change between consecutive points is constant. Since all of the points have $x -$ and $y -$ coordinates that are identifiable, we can use slope triangles to determine if this is the case.

For each $1$ unit we move in the positive horizontal direction, we move $3$ units in the negative vertical direction. Therefore, the sequence is arithmetic with a common difference of $- 3 .$

Graph 2

The graph decreases up to $n = 4,$ and then increases after. This means the sequence cannot be arithmetic. But is it geometric? A geometric sequence will either decrease or increase forever with a common factor. Therefore, since the graph first decreases and then increases, it cannot be geometric either. Therefore, it is neither.

Graph $3$

As we already discussed, a geometric sequence either increases or decreases forever with a common factor. Examining the graph, the sequence does exhibit the characteristics of a decreasing geometric sequence. However, to be sure we will record the coordinates of the points where it is possible.

The coordinates of the remaining points we cannot be sure of. However, we will assume they follow the geometric sequence that the first five points do as well. Let's investigate this using a rate-of-change table.

As we can see, the sequence is geometric with a common ratio of $\frac{1}{2} .$

Graph 1

Graph 2

Graph 3

Graph $3$

Exercise