Arithmetic Sequences

Terms of an Arithmetic Sequence

Learning Outcomes

• An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.

• This constant is called the common difference.

• Values change by a constant amount each year, where each term increases or decreases by the same constant value.

• For example, the common difference is –3,400.

• You can choose any term of the sequence and add 3 to find the subsequent term.

• The learning outcomes are to find the common difference for an arithmetic sequence and to find terms of an arithmetic sequence.

Arithmetic Sequence

• If $a1$ is the first term of an arithmetic sequence and $d$ is the common difference, the sequence will be: $a_n = a_1 + (n-1)d, \quad n \ge 1$

Finding Common Differences

• To determine whether a common difference exists, subtract each term from the subsequent term.

Example:

1. \left{1, 2, 4, 8, 16,… \right} is not arithmetic because there is no common difference.

2. \left{-3, 1, 5, 9, 13,… \right} is arithmetic because there is a common difference of 4.

   • $1 - (-3) = 4$

   • $5 - 1 = 4$

   • $9 - 5 = 4$

   • $13 - 9 = 4$

Analysis of the Solution

• Arithmetic sequences have a constant rate of change, so their graphs will always be points on a line.

Q & A

• If we are told that a sequence is arithmetic, we can choose any one term in the sequence and subtract it from the subsequent term to find the common difference.

Try It

• Is \left{18, 16, 14, 12, 10, … \right} arithmetic? If so, find the common difference.

  • Yes, the common difference is -2.

• Is \left{1, 3, 6, 10, 15,… \right} arithmetic?

  • No, because $3 - 1 \neq 6 - 3$

Writing Terms of Arithmetic Sequences

• To find terms, begin with the first term and add the common difference repeatedly.

• Any term can be found by plugging in the values of $a<em>1$, $n$, and $d$ into the formula: $an = a_1 + (n - 1)d$

How To:

1. Add the common difference to the first term to find the second term.

2. Add the common difference to the second term to find the third term.

3. Continue until all of the desired terms are identified.

4. Write the terms separated by commas within brackets.

Example:

• Write the first five terms of the arithmetic sequence with $a_1 = 17$ and $d = -3$

  • The first five terms are \left{17, 14, 11, 8, 5 \right}

Analysis of the Solution

• The graph of the sequence consists of points on a line.

Try It

• List the first five terms of the arithmetic sequence with $a_1 = 1$ and $d = 5$

  • \left{1, 6, 11, 16, 21 \right}

How To:

• Given the first term and any other term in an arithmetic sequence, find a given term.

1. Substitute the values given for $a<em>1$, $a</em>n$, and $n$ into the formula $a<em>n = a</em>1 + (n - 1)d$ to solve for $d$.

2. Find a given term by substituting the appropriate values for $a<em>1$, $n$, and $d$ into the formula $a</em>n = a_1 + (n - 1)d$

Example:

• Given $a<em>1 = 8$ and $a</em>4 = 14$, find $a_5$

  • The sequence can be written in terms of the initial term 8 and the common difference $d$: \left{8, 8+d, 8+2d, 8+3d \right}

  • We know the fourth term equals 14; we know the fourth term has the form $a_1 + 3d = 8 + 3d$

  • We can find the common difference $d$

  • $a<em>n = a</em>1 + (n - 1)d$

  • $a<em>4 = a</em>1 + 3d$

  • $a_4 = 8 + 3d$

  • Write the fourth term of the sequence in terms of $a_1$ and $d$

  • $14 = 8 + 3d$

  • Substitute 14 for $a_4$

  • $d = 2$

  • Solve for the common difference

  • Find the fifth term by adding the common difference to the fourth term.

  • $a<em>5 = a</em>4 + 2 = 16$

Analysis of the Solution

• The common difference is added to the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on.

• The tenth term could be found by adding the common difference to the first term nine times or by using the equation $a<em>n = a</em>1 + (n - 1)d$

Try It

• Given $a<em>3 = 7$ and $a</em>5 = 17$, find $a_2$