An arithmetic sequence is a sequence of numbers in which each successive term increases or decreases by a constant value, called the common difference.
A geometric sequence is one in which each successive term of the series is obtained by multiplying the previous term by a constant value called the common ratio.
In the series below:
[tex]\mleft\lbrace a_n\mright\rbrace\text{ = }\mleft\lbrace1,\text{ 24, 5, 28,}\ldots\mright\rbrace[/tex]first term = 1
second term = 24
third term = 5
fourth term = 28
Common difference d:
The difference between the second and first terms must be equal to the difference between the third and the second term.
[tex]\begin{gathered} d\text{ = second term - first term} \\ =\text{ 24 -1} \\ d\text{ = 23} \end{gathered}[/tex][tex]\begin{gathered} d\text{ = third term - second term} \\ =5-24 \\ d\text{ = -19} \end{gathered}[/tex]Since the common differences obtained above are not equal, the sequence is thus not an arithmetic sequence.
Common ratio r:
The common ratio between the second and the first term must be equal to the common ratio between the third and the second term.
[tex]\begin{gathered} r\text{ = }\frac{\sec ond\text{ term}}{first\text{ term}} \\ =\frac{24}{1} \\ \Rightarrow r=24 \end{gathered}[/tex][tex]\begin{gathered} r=\frac{third\text{ term}}{\sec ond\text{ term}} \\ =\frac{5}{24} \\ \Rightarrow r=0.208 \end{gathered}[/tex]Since the common ratios obtained above are not equal, the sequence is thus not a geometric sequence.
Hence, the sequence is neither an arithmetic sequence nor a geometric sequence.
The correct option is C.