Writing Terms of Arithmetic Sequences

Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of \(n\) and \(d\) into formula below.

\(a_n=a_1+(n−1)d\)


How To

Given the first term and the common difference of an arithmetic sequence, find the first several terms.

  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 2

Writing Terms of Arithmetic Sequences

Write the first five terms of the arithmetic sequence with \(a_1=17\) and \(d=−3\).


Solution

Adding \(−3\) is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.

The first five terms are \(\{17,14,11,8,5\}\)


Analysis

As expected, the graph of the sequence consists of points on a line as shown in Figure 2.

Figure 2


Try It #3

List the first five terms of the arithmetic sequence with \(a_1=1\) and \(d=5\).


How To

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

  1. Substitute the values given for \(a_1,a_n,n\) into the formula \(a_n=a_1+(n−1)d\) to solve for \(d\).
  2. Find a given term by substituting the appropriate values for \(a_1,n\), and \(d\) into the formula \(a_n=a_1+(n−1)d\).


Example 3

Writing Terms of Arithmetic Sequences

Given \(a_1=8\) and \(a_4=14\), find \(a_5\).


Solution

The sequence can be written in terms of the initial term 8 and the common difference \(d\).

\(\{8,8+d,8+2d,8+3d\}\)

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 \(d\).

\(\begin{array}{ll}
a_{n}=a_{1}+(n-1) d & \\
a_{4}=a_{1}+3 d & \\
a_{4}=8+3 d & \text { Write the four th term of the sequence in terms of } a_{1} \text { and } d . \\
14=8+3 d & \text { Substitute } 14 \text { for } a_{4} . \\
d=2 & \text { Solve for the common difference. }
\end{array}\)

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

\(a_5=a_4+2=16\)


Analysis

Notice that 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 \(a_n=a_1+(n−1)d\).


Try It #4

Given \(a_3=7\) and \(a_5=17\), find \(a_2\).