To determine whether the situation represents an arithmetic sequence or a geometric sequence, let's define these two types of sequences and then apply the definitions to the situation described.
An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. This constant difference is also known as the common difference. The general form of an arithmetic sequence is [tex]\(a, a+d, a+2d, a+3d, \ldots\)[/tex] where [tex]\(a\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
A geometric sequence, on the other hand, is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant ratio, known as the common ratio. The general form of a geometric sequence is [tex]\(a, ar, ar^2, ar^3, \ldots\)[/tex] where [tex]\(a\)[/tex] is the first term and [tex]\(r\)[/tex] is the common ratio.
Now, let's analyze the situation described:
You have a pea plant that grows 0.5 inches every day. If the plant starts at a certain height, each day, it will increase in height by 0.5 inches.
For example, if the initial height of the plant is [tex]\(h\)[/tex] inches, the growth would look like this:
- Day 1: [tex]\(h + 0.5\)[/tex]
- Day 2: [tex]\((h + 0.5) + 0.5 = h + 1.0\)[/tex]
- Day 3: [tex]\((h + 1.0) + 0.5 = h + 1.5\)[/tex]
- and so on...
As we can see, each day, the height of the pea plant increases by 0.5 inches from the height of the previous day. This is a constant addition of 0.5 inches, and there is no multiplication involved.
Since the growth of the pea plant is represented by a fixed amount (0.5 inches) being added each day to the previous day's height, this situation is an example of an arithmetic sequence with the common difference [tex]\(d\)[/tex] being 0.5 inches.
Therefore, the growth of the pea plant as it grows 0.5 inches every day would be represented by an arithmetic sequence.
