Use a Formula for an Arithmetic Sequence

How To

Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

1. Find the common difference $d$.

2. Substitute the common difference and the first term into $a_n=a_1+d(n–1)$.

3. Substitute the last term for $a_n$ and solve for $n$.

Example 6

Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence.

$\{8, 1, –6, ..., –41\}$

Solution

The common difference can be found by subtracting the first term from the second term.

$1−8=−7$

The common difference is$−7$. Substitute the common difference and the initial term of the sequence into the $nth$ term formula and simplify.

$\begin{array}{ll}a_n = a_1+d(n-1) \\a_n = 8+(-7)(n-1) \\a_n= 15 - 7n\end{array}$

Substitute $−41$ for $a_n$ and solve for $n$

$\begin{array}{ll}-41 &= 15 -7n \\8 &= n\end{array}$

There are eight terms in the sequence.