Arithmetic Sequences

An arithmetic sequence changes by a constant difference and can be described recursively or with a linear explicit formula.

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A sequence tracks terms; a series tracks their accumulating partial sums. Indexed terms appear along a horizontal progression above partial-sum markers. Shrinking term arrows and a target band distinguish terms approaching zero from partial sums actually converging.

Arithmetic sequences are linear patterns whose consecutive terms differ by the same amount.

Sequence vocabulary

A sequence is an ordered list of terms. The notation $t_n$ or $a_n$ means the term in position $n$. A finite sequence ends; an infinite sequence continues according to a rule.

Order matters. The values $4,7,10,13,\ldots$ form a different object from the set containing those values because each term has an assigned position.

Recursive form

A recursive rule states a starting term and explains how to obtain the next term:

t_1=a,qquad t_n=t_{n-1}+d\quad(n\ge2).

For $4,7,10,\ldots$,

t_1=4,qquad t_n=t_{n-1}+3.

Recursive form mirrors the pattern but requires earlier terms to reach a distant term.

Worked example: find a distant term

Finding a missing position

For the sequence $7,11,15,\ldots$, the explicit rule is

$$ t_n=7+4(n-1)=4n+3. $$

To find which term equals $83$, solve

$$ 4n+3=83, $$

so $n=20$. Since a term number must be a positive integer, a non-integer solution would mean $83$ is not a term of the sequence.

Building a rule from two terms

Suppose $t_5=17$ and $t_{12}=45$. Between positions $5$ and $12$ are $7$ equal steps:

d=\frac{45-17}{12-5}=4.

Then $17=a+4(5-1)$, so $a=1$. The sequence is $t_n=1+4(n-1)=4n-3$.

This is the same slope reasoning used for a line through two points.

Common mistakes

Using $nd$ instead of $(n-1)d$. No difference is added before the first term.

Finding a difference in the wrong order. Use next term minus previous term consistently.

Confusing a sequence with its sum. $t_n$ is one term; $S_n$ is a total of terms.

Drawing a continuous line without context. Sequence inputs are normally whole-number positions.

Accepting a non-integer term number. Positions must belong to the stated domain.