The Solution:
Given the sequence below:
[tex]26,18,10,2,\ldots[/tex]We are required to identify the type of sequence and to find the explicit rule for the given sequence.
Step 1:
To determine the type of sequence, we shall run the following check:
[tex]\begin{gathered} a_2-a_1=a_3-a_2\text{ then it is an Arithmetic sequence} \\ \text{ but if} \\ \frac{a_2}{a_1}=\frac{a_3}{a_2}\text{ then it is a Geometric sequence.} \end{gathered}[/tex]So,
[tex]\begin{gathered} 18-26=10-18 \\ -8=-8 \\ \text{ Then it follows that the sequence is an Arithmetic sequence.} \end{gathered}[/tex]Thus, the sequence is an arithmetic sequence.
Step 2:
To find the explicit rule, we shall use the formula below:
[tex]a_n=a_1+(n-1)d[/tex]In this case,
[tex]\begin{gathered} a_1=\text{ first term=26} \\ n=\text{ number of terms=?} \\ d=\text{ co}mmon\text{ difference=18-26=-8} \end{gathered}[/tex]Substituting these values in the formula, we get
[tex]\begin{gathered} a_n=26+(n-1)(-8) \\ a_n=26-8(n-1) \end{gathered}[/tex]Therefore, the explicit rule of the sequence is :
[tex]a_n=26-8(n-1)[/tex]