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An arithmetic sequence, or arithmetic progression, is a set of numbers in which the difference between consecutive terms (terms that come after one another) is constant. This difference is called the common difference. For example, all of the consecutive terms in the arithmetic sequence:
$$1,4,7,10,13,16,19,...$$
share a common difference of $$3$$.
Note: The three dots (. . .) mean that this sequence is infinite.

Though others can also be used, the following variables are typically used to represent the terms of an arithmetic sequence:
$$a_1$$ represents the first term of the sequence. In the example above, $$a_1=1$$
$$a_n$$ represents the nth term (a term we are trying to find).
$$d$$ represents the common difference between consecutive terms. In the example above, $$d=3$$
$$n$$ represents the number of terms in the sequence. In the example above, $$n=7$$

The standard form of arithmetic sequences can be expressed as: $$a,a+d,a+2d,a+3d,a+4d,a+5d...$$
$$a$$ represents the first term and is sometimes written as $$a_1$$.
$$d$$ represents the common difference.

Formulas

Finding any term ($$a_n$$) in an arithmetic sequence:
$$a_n=a+d\left(n-1\right)$$

$$a$$ represents the first term.
$$d$$ represents the common difference.
$$n$$ represents the position of a term in the sequence.
A sequence with $$n$$ number of terms would be written as:
$$a,a+d\left(2-1\right),a+d\left(3-1\right),a+d\left(4-1\right),a+d\left(5-1\right),a+d\left(6-1\right)...a+d\left(n-1\right)$$
in which the last term's common difference is multiplied by $$n-1$$ (because $$d$$ is not used in the 1st term).

Example: To find the next term in:
$$1,4,7,10,13,16,19...$$
which would be the 8th term, we would plug the following into the general term formula $$a_n=a+d\left(n-1\right)$$:
$$a$$ (first term) $$=1$$
$$d$$ (common difference) $$=3$$
$$n$$ (term number) $$=8$$
This would give us:
$$a_8=1+3\left(8-1\right)$$ which we could solve to get $$a_8=22$$.
So, our sequence would be: $$1,4,7,10,13,16,19,22...$$

Finding the sum of all the terms in an arithmetic sequence:
$$s=n\left(a_1+a_n\right)/2$$

$$s$$ is the sum of the terms in the sequence.
$$a$$ represents the first term.
$$n$$ represents the position of a term in the sequence.
$$d$$ represents the common difference.
Example: To find the sum of:
$$1,4,7,10,13,16,19...$$ we plug the following into the sum formula $$s=n\left(a_1+a_n\right)/2$$ :
$$n$$ (total number of terms)$$=7$$
$$a$$ (first term)$$=1$$
$$a_n$$ (the last term)$$=19$$
This would give us:
$$s=7\left(1+19\right)/2$$ which we could solve to get $$s=70$$.
So, the sum of the sequence would be: $$70$$
Tiger identifies arithmetic sequences and displays their terms, the sum of their terms, and their explicit and recursive forms.