The given sequence is:
[tex]9,-18,27,-36,\ldots[/tex]
It is required to find the nth term of the sequence as suggested by the pattern.
Rewrite the terms of the sequence as follows:
[tex]\begin{gathered} 9\times1,9\times-2,9\times3,9\times-4,\ldots \\ \text{ Rewrite as follows to denote the alternating terms:} \\ 9\times(-1)^2\times1,9\times(-1)^3\times2,9\times(-1)^4\times3,9\times(-1)^5\times4,\operatorname{\ldots} \end{gathered}[/tex]
The powers can be written as:
[tex]9\times(-1)^{1+1}\times1,9\times(-1)^{2+1}\times2,9\times(-1)^{(3+1)}\times3,9\times(-1)^{(4+1)}\times4,\ldots[/tex]
From the pattern above, it follows that the nth term of the sequence is:
[tex]\begin{gathered} \lbrace a_n\rbrace=\lbrace9\times(-1)^{n+1}\times n\rbrace \\ \Rightarrow\lbrace a_n\rbrace=\lbrace(-1)^{(n+1)}\cdot9n\rbrace \end{gathered}[/tex]
The nth term of the sequence is shown above.
