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Study Guides > College Algebra

Formulas for Arithmetic Sequences

Learning Objectives

  • Write an explicit formula for an arithmetic sequence
  • Write a recursive formula for the arithmetic sequence
 

Using Explicit Formulas for Arithmetic Sequences

We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.

an=a1+d(n1){a}_{n}={a}_{1}+d\left(n - 1\right)

To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence. A sequence, {200, 150, 100, 50, 0, ...}, that shows the terms differ only by -50. The common difference is 50-50 , so the sequence represents a linear function with a slope of 50-50 . To find the yy -intercept, we subtract 50-50 from 200:200(50)=200+50=250200:200-\left(-50\right)=200+50=250 . You can also find the yy -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. Graph of the arithmetic sequence. The points form a negative line. Recall the slope-intercept form of a line is y=mx+by=mx+b. When dealing with sequences, we use an{a}_{n} in place of yy and nn in place of xx. If we know the slope and vertical intercept of the function, we can substitute them for mm and bb in the slope-intercept form of a line. Substituting 50-50 for the slope and 250250 for the vertical intercept, we get the following equation:

an=50n+250{a}_{n}=-50n+250

We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is an=20050(n1){a}_{n}=200 - 50\left(n - 1\right) , which simplifies to an=50n+250{a}_{n}=-50n+250.

A General Note: Explicit Formula for an Arithmetic Sequence

An explicit formula for the nthn\text{th} term of an arithmetic sequence is given by

an=a1+d(n1){a}_{n}={a}_{1}+d\left(n - 1\right)

How To: Given the first several terms for an arithmetic sequence, write an explicit formula.

  1. Find the common difference, a2a1{a}_{2}-{a}_{1}.
  2. Substitute the common difference and the first term into an=a1+d(n1){a}_{n}={a}_{1}+d\left(n - 1\right).

Example: Writing the nth Term Explicit Formula for an Arithmetic Sequence

Write an explicit formula for the arithmetic sequence. {212223242}\left\{2\text{, }12\text{, }22\text{, }32\text{, }42\text{, }\ldots \right\}

Answer: The common difference can be found by subtracting the first term from the second term.

d=a2a1=122=10\begin{array}{ll}d\hfill & ={a}_{2}-{a}_{1}\hfill \\ \hfill & =12 - 2\hfill \\ \hfill & =10\hfill \end{array}

The common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.

an=2+10(n1)an=10n8\begin{array}{l}{a}_{n}=2+10\left(n - 1\right)\hfill \\ {a}_{n}=10n - 8\hfill \end{array}

Analysis of the Solution

The graph of this sequence shows a slope of 10 and a vertical intercept of 8-8 . Graph of the arithmetic sequence. The points form a positive line.

Try It

Write an explicit formula for the following arithmetic sequence. {50,47,44,41,}\left\{50,47,44,41,\dots \right\}

Answer: an=533n{a}_{n}=53 - 3n

Find the Number of Terms in an Arithmetic Sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

  1. Find the common difference dd.
  2. Substitute the common difference and the first term into an=a1+d(n1){a}_{n}={a}_{1}+d\left(n - 1\right).
  3. Substitute the last term for an{a}_{n} and solve for nn.

Example: Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence. {816...41}\left\{8\text{, }1\text{, }-6\text{, }...\text{, }-41\right\}

Answer: The common difference can be found by subtracting the first term from the second term.

18=71 - 8=-7

The common difference is 7-7 . Substitute the common difference and the initial term of the sequence into the nthn\text{th} term formula and simplify.

an=a1+d(n1)an=8+7(n1)an=157n\begin{array}{l}{a}_{n}={a}_{1}+d\left(n - 1\right)\hfill \\ {a}_{n}=8+-7\left(n - 1\right)\hfill \\ {a}_{n}=15 - 7n\hfill \end{array}

Substitute 41-41 for an{a}_{n} and solve for nn

41=157n8=n\begin{array}{l}-41=15 - 7n\hfill \\ 8=n\hfill \end{array}

There are eight terms in the sequence.

Try It

Find the number of terms in the finite arithmetic sequence. {61116...56}\left\{6\text{, }11\text{, }16\text{, }...\text{, }56\right\}

Answer: There are 11 terms in the sequence.

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