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Algebra Sequences (as functions)

Arithmetic sequences

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  • Arithmetic Sequence (Explicit Formula) Watch
  • Explicit and recursive definitions of sequences | Precalculus | Khan Academy Watch
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Theory

An arithmetic sequence adds a constant common difference \(d\) each step:

\[a_n=a_1+(n-1)d.\]

It behaves like a linear function of the term number \(n\).

The common difference is the slope of the sequence.
An arithmetic sequence Arithmetic terms increase by a constant difference, so they lie on a line. n +3 each step
Terms rise by \(d=3\) each step.
Arithmetic sequence Arithmetic sequence Arithmetic sequence add a common difference d nth term: aₙ = a₁ + (n-1)d like a linear function
The arithmetic rule.

nth term:

\[a_n=a_1+(n-1)d\]
a sub n equals a one plus n minus 1 times d
Find \(d\) by subtracting consecutive terms.

How to work with arithmetic sequences

  1. Find the common difference \(d\).
  2. Identify the first term \(a_1\).
  3. Use \(a_n=a_1+(n-1)d\).
  4. Substitute the term number \(n\).
Example 1 — Common difference
Find the common difference of \(2,5,8,11,\dots\).
Solution

Subtract consecutive terms.

\(5-2\)\(=\)\(3\)
the common difference is 3
Example 2 — nth term
Find the \(10\)th term of \(2,5,8,\dots\).
Solution

Use \(a_n=a_1+(n-1)d\) with \(a_1=2,d=3\).

\(a_{10}\)\(=\)\(2+9(3)=29\)
the tenth term is 29
Example 3 — Explicit rule
Write an explicit rule for \(2,5,8,\dots\).
Solution

Simplify \(a_1+(n-1)d\).

\(a_n\)\(=\)\(2+(n-1)3=3n-1\)
a sub n equals 3 n minus 1
Example 4 — Recursive rule
Write a recursive rule for \(2,5,8,\dots\).
Solution

Each term adds \(3\).

\(a_1=2,\)\(a_n=a_{n-1}+3\)
a one is 2 and each term adds 3

Common pitfalls

Use \((n-1)\), not \(n\), in the formula.
A common difference is added, not multiplied.
Check \(d\) is truly constant.

Frequently asked questions

What is an arithmetic sequence?

A sequence with a constant difference between terms.

What is the nth term formula?

\(a_n=a_1+(n-1)d\).

How do you find the common difference?

Subtract any term from the next.

How is it like a linear function?

The common difference acts as the slope.

More in Sequences (as functions)
Geometric sequences
20 questions · Video