Arithmetic Sequences (College Board AP® Precalculus): Revision Note

Arithmetic sequences

What is an arithmetic sequence?

• An arithmetic sequence is a sequence in which successive terms have a common difference

  • This means the same value is added to each term to get the next term

    • This common difference is also described as a constant rate of change

  • The common difference is usually denoted by $d$

• E.g. the sequence $3,7,11,15,19,\dots$ is arithmetic

  • with a common difference of $d=4$

  • Each term is 4 more than the previous term

    • $7-3=4$, $11-7=4$, $15-11=4$, etc.

• The common difference of a sequence can be

  • positive (the sequence increases)

  • negative (the sequence decreases)

  • or zero (all terms are the same)

How can I find the common difference of an arithmetic sequence?

• To find the common difference of an arithmetic sequence

  • Subtract any term from the term after it

    • $d=a_{n+1}-a_n$

• If the differences between consecutive terms are not all equal, then the sequence is not arithmetic

What is the general term of an arithmetic sequence?

• The general term (also called the nth term) of an arithmetic sequence can be written in two forms

• Using the initial value

  • $a_n=a_0+dn$

    • where $a_0$ is the initial value (the term when $n=0$)

    • and $d$ is the common difference

• Using any known term

  • $a_n=a_k+d(n-k)$

    • where $a_k$ is the value of the known $k$th term

    • and $d$ is the common difference

  • This form is useful when you don't know $a_0$ but you do know a different term in the sequence

• Both forms express the same idea

  • Start from a known term and add the common difference the appropriate number of times

How do these formulas work in practice?

• E.g. an arithmetic sequence has $a_0=5$ and a common difference of $d=3$

  • The general term is $a_n=5+3n$

  • So: $a_0=5$, $a_1=8$, $a_2=11$, $a_3=14$, $\dots$

• Or e.g. you are told that $a_4=20$ and $d=-2$

  • Using the second form: $a_n=20+(-2)(n-4)=20-2n+8=28-2n$

  • You can verify that to check the answer

    • $a_4=28-2\left(4\right)=28-8=20$ ✓

What does the graph of an arithmetic sequence look like?

• Because an arithmetic sequence is a function of the whole numbers

  • its graph is a set of discrete points

  • as discussed in the Sequence Basics study guide

• The points in the graph of an arithmetic sequence lie along a straight line

  • This is because the common difference $d$ acts like a constant slope

  • The general term $a_n=a_0+dn$

    • has the same structure as a linear function $y=b+mx$

• However, the graph is not a continuous line

  • it is only the individual points at whole number values of $n$