Question

write an explicit formula for $a_n$, the $n^{\text{th}}$ term of the sequence $15, 7, -1, ....$

Explanation:

Step1: Identify the sequence type

The sequence \(15, 7, -1, \dots\) is an arithmetic sequence because the difference between consecutive terms is constant. Let's check the common difference \(d\):
\(d = 7 - 15=-8\), and \(-1 - 7=-8\), so \(d = - 8\).

Step2: Recall the arithmetic sequence formula

The explicit formula for an arithmetic sequence is \(a_{n}=a_{1}+(n - 1)d\), where \(a_{1}\) is the first term and \(d\) is the common difference.

Step3: Substitute the values

Here, \(a_{1}=15\) and \(d=-8\). Substituting these into the formula:
\[

$$\begin{align*} a_{n}&=15+(n - 1)(-8)\\ &=15-8n + 8\\ &=23-8n \end{align*}$$

\]

Answer:

\(a_{n}=23 - 8n\)