What is an Arithmetic Sequence?


Sequences

Previously, we were introduced to sequences.

Recall

We normally identify terms in a sequence using the letter \(t\) with a subscript indicating the term number.

For example, in the sequence \(J,~F,~M,~A,~M,~J,~J,~A,~S,~O,~N,~D\):

  • \(t_2=F\),
  • \(t_{12}=D\), and
  • \(t_n\) can be described as the first letter of the \(n\)th month of the year. 

In this lesson, we will consider a class of numerical sequences called arithmetic sequences.


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Arithmetic Sequences

An arithmetic sequence is a sequence in which each term after the first term is obtained by adding a constant \(d\) to the previous term.

 

 

Example 1

An arithmetic sequence has first term \(5\) and common difference \(-3\).


 

 

Check Your Understanding 1

An arithmetic sequence has first term \(((((t)*(e))*(r))*(m))*1.0\) and common difference \((((d)*(v))*(a))*(l)\). Determine the first three terms of the sequence.

Enter the first three terms in order, separated by commas.  Example:  3,4,5

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Since the common difference is \(d=(((d)*(v))*(a))*(l)\), each term is \((((abs(D))*(v))*(a))*(l)\) (((((((((((m)*(o))*(r))*(e))*(L))*(e))*(s))*(s))*(W))*(o))*(r))*(d) than the previous term.

Therefore, the first three terms listed in order are \(((((t)*(e))*(r))*(m))*1.0,~((((t)*(e))*(r))*(m))*2.0,~ ((((t)*(e))*(r))*(m))*3.0\).

A Recursive Definition

The previous example hinted at a way that we can define an arithmetic sequence recursively.

 

 

Example 2

For the arithmetic sequence \(\dfrac7{12},~\dfrac{5}{6},~\dfrac {13}{12},\ldots \) determine

 

 

Example 2 — Part A

For the arithmetic sequence \(\dfrac7{12},~\dfrac56,~\dfrac {13}{12},\ldots \) determine

  1. The common difference, \(d\).
  2. A recursive formula for the sequence.

 

 

Example 2 — Part B

For the arithmetic sequence \(\dfrac7{12},~\dfrac56,~\dfrac {13}{12},\ldots \) determine

  1. The common difference, \(d\).
  2. A recursive formula for the sequence.

 

 

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Check Your Understanding 2


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Example 3

Identify whether or not each sequence is arithmetic. For the sequences that are arithmetic, determine the common ratio, \(d\).

  1. \(1,~4,~9,~16\)
  2. \(191,~195,~199,~203\)
  3. \(\dfrac83,~2,~\dfrac43,~\dfrac23\)

Consider each pair of consecutive terms. If the sequence is arithmetic, the differences of consecutive pairs of terms, \(t_n-t_{n-1}\), will be constant.

Solution — Part A

 \(1,~4,~9,~16\)

The differences of consecutive pairs of terms are:

  • \(t_2-t_1=4-1=3\),
  • \(t_3-t_2=9-4=5\), 
  • \(t_4-t_3=16-9=7\).

Since the differences are not all the same, the sequence does not have a common difference. Therefore, the sequence \(1,~4,~9,~16\) is not arithmetic.

Solution — Part B

 \(191,~195,~199,~203\)

The differences of consecutive pairs of terms are:

  • \(t_2-t_1=195-191=4\),
  • \(t_3-t_2=199-195=4\),
  • \(t_4-t_3=203-199=4\).

Since the differences are all the same, the sequence has a common difference. Therefore, the sequence \(191, 195, 199, 203\) is arithmetic with \(d=4\).

Solution — Part C

  \(\dfrac83,~2,~\dfrac43,~\dfrac23\)

The differences of consecutive pairs of terms are:

  • \(t_2-t_1=2-\dfrac83=-\dfrac23\),
  • \(t_3-t_2=\dfrac43-2=-\dfrac23\),
  • \(t_4-t_3=\dfrac23-\dfrac43=-\dfrac23\).

Since the differences are all the same, the sequence has a common difference.
Therefore, the sequence \(\dfrac83,~2,~\dfrac43,~\dfrac23\)  is arithmetic with \(d=-\dfrac23\).


Check Your Understanding 3


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