Previously, we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is [latex]a_1[/latex] and the common difference [latex]d[/latex]. We will then look for a pattern.
Did you notice that the number of [latex]d[/latex]s that were added to [latex]a_1[/latex] is one less than the number of the term?
general term (nth term) of an arithmetic sequence
The general term of an arithmetic sequence with first term [latex]a_1[/latex] and the common difference [latex]d[/latex] is
Write the first five terms of the arithmetic sequence with [latex]{a}_{1}=17[/latex] and [latex]d=-3[/latex].
Adding [latex]-3[/latex] is the same as subtracting [latex]3[/latex]. Beginning with the first term, subtract [latex]3[/latex] from each term to find the next term.The first five terms are [latex]\left\{17,14,11,8,5\right\}[/latex]
Analysis of the Solution
As expected, the graph of the sequence consists of points on a line.
How To: Given any the first term and any other term in an arithmetic sequence, find a given term.
Substitute the values given for [latex]{a}_{1},{a}_{n},n[/latex] into the formula [latex]{a}_{n}={a}_{1}+\left(n - 1\right)d[/latex] to solve for [latex]d[/latex].
Find a given term by substituting the appropriate values for [latex]{a}_{1},n[/latex], and [latex]d[/latex] into the formula [latex]{a}_{n}={a}_{1}+\left(n - 1\right)d[/latex].
Given [latex]{a}_{1}=8[/latex] and [latex]{a}_{4}=14[/latex] , find [latex]{a}_{5}[/latex] .
The sequence can be written in terms of the initial term [latex]8[/latex] and the common difference [latex]d[/latex] .
[latex]\left\{8,8+d,8+2d,8+3d\right\}[/latex]
We know the fourth term equals [latex]14[/latex]; we know the fourth term has the form [latex]{a}_{1}+3d=8+3d[/latex] .
We can find the common difference [latex]d[/latex] .
[latex]\begin{align}&{a}_{n}={a}_{1}+\left(n - 1\right)d \\ &{a}_{4}={a}_{1}+3d \\ &{a}_{4}=8+3d && \text{Write the fourth term of the sequence in terms of } {a}_{1} \text{ and } d. \\ &14=8+3d && \text{Substitute } 14 \text{ for } {a}_{4}. \\ &d=2 && \text{Solve for the common difference}. \end{align}[/latex]
Find the fifth term by adding the common difference to the fourth term.
[latex]{a}_{5}={a}_{4}+2=16[/latex]
Analysis of the Solution
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 [latex]{a}_{n}={a}_{1}+\left(n - 1\right)d[/latex].
