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.
- Find the common difference [latex]d[/latex].
- Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex].
- Substitute the last term for [latex]{a}_{n}[/latex] and solve for [latex]n[/latex].
[latex]\left\{8,1,-6, \dots ,-41\right\}[/latex]
- The common difference can be found by subtracting the first term from the second term
[latex]1 - 8=-7[/latex]
The common difference is [latex]-7[/latex]. - Substitute the common difference and the initial term of the sequence into the [latex]n\text{th}[/latex] term formula and simplify.[latex]\begin{align}&{a}_{n}={a}_{1}+d\left(n - 1\right) \\ &{a}_{n}=8+-7\left(n - 1\right) \\ &{a}_{n}=15 - 7n \end{align}[/latex]
- Substitute [latex]-41[/latex] for [latex]{a}_{n}[/latex] and solve for [latex]n[/latex]
[latex]\begin{align}-41&=15 - 7n \\ 8&=n \end{align}[/latex]
This means that there are eight terms in the sequence.