Arithmetic Sequences

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.

Constant first difference

An arithmetic sequence has a constant common difference $d$:

For $4,7,10,13,\ldots$, each term is $3$ more than the previous term, so $d=3$. A negative difference produces a decreasing sequence, while $d=0$ produces a constant sequence.

Recursive form

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

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

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

Explicit form

From term $1$ to term $n$, the common difference is added $n-1$ times. Therefore

For $a=4$ and $d=3$,

The explicit rule gives any term directly.

Worked example: find a distant term

Finding a missing position

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

To find which term equals $83$, solve

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:

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.

Connection to linear functions

The graph of ordered pairs $(n,t_n)$ lies on a line with slope $d$. However, a sequence normally has domain $n=1,2,3,\ldots$, so its graph consists of separate points rather than a continuous line.

An arithmetic sequence is a discrete linear relation. Its common difference corresponds to slope, and its explicit formula resembles slope-intercept form.

Modelling applications

Arithmetic sequences model situations with equal additive change: rows of seats increasing by a fixed count, a tank losing the same volume each minute, or a savings plan adding the same deposit each month without interest.

Check whether the constant-change assumption and discrete domain fit the context. A term count cannot usually be fractional or negative.

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.

Quick self-check

• Are consecutive first differences constant?
• What are $a$ and $d$, including their signs?
• Did I use $n-1$ differences to reach term $n$?
• Does the explicit rule reproduce the first few terms?
• Is the requested position a valid positive integer?

Related topics

• Geometric Sequences
• Series
• Linear Function
• Slope