What is the difference between arithmetic and geometric sequences?

Step 1. Arithmetic Sequence:
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference ($d$). The general form of an arithmetic sequence is given by the formula:
$a_n=a_1+(n-1)d$
where $a_n$ represents the $n$th term, $a_1$ is the first term, $n$ is the position of the term, and $d$ is the common difference.
Step 2. Geometric Sequence:
A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the preceding term by a constant ratio. This constant ratio is called the common ratio ($r$). The general form of a geometric sequence is given by the formula:
$a_n=a_1·r^{(n-1)}$ where $a_n$ represents the $n$th term, $a_1$ is the first term, $n$ is the position of the term, and $r$ is the common ratio.
In summary, arithmetic sequences have a constant difference between terms, while geometric sequences have a constant ratio between terms.

The difference between arithmetic and geometric sequences can be summarized as follows:
1. Arithmetic Sequence:
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same. This common difference is denoted by $d$. In an arithmetic sequence, each term can be found by adding the common difference to the previous term.
The general form of an arithmetic sequence is given by:
$a_1,a_1+d,a_1+2d,a_1+3d,\dots$
where $a_1$ represents the first term of the sequence.
2. Geometric Sequence:
A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant ratio. This common ratio is denoted by $r$. In a geometric sequence, the ratio of any term to its previous term remains constant.
The general form of a geometric sequence is given by:
$a_1,a_1\times r,a_1\times r^2,a_1\times r^3,\dots$
where $a_1$ represents the first term of the sequence.
To summarize the differences:
- Arithmetic sequences have a constant difference between consecutive terms ($d$), whereas geometric sequences have a constant ratio between consecutive terms ($r$).
- In an arithmetic sequence, each term is obtained by adding the common difference to the previous term, while in a geometric sequence, each term is obtained by multiplying the previous term by the common ratio.
- The general form of an arithmetic sequence is $a_1,a_1+d,a_1+2d,a_1+3d,\dots$, whereas the general form of a geometric sequence is $a_1,a_1\times r,a_1\times r^2,a_1\times r^3,\dots$.
Remember that $a_1$ represents the first term of the sequence, $d$ represents the common difference in an arithmetic sequence, and $r$ represents the common ratio in a geometric sequence.