Answer :
Notice that the sequence of the absolute values of the given sequence is just the sequence of the natural numbers:
[tex]1,2,3,4,5,\ldots[/tex]Nevertheless, the sign of each member of the sequence alternates between positive and negative:
[tex]+,-,+,-,+,\ldots[/tex]We can explicitly represent the n-th term of the sequence as:
[tex]a_n=n\times(-1)^{n-1}[/tex]Verify this expression for the first 3 terms:
[tex]\begin{gathered} a_1=1\times(-1)^{1-1}=1\times(-1)^0=1\times1=1 \\ a_2=2\times(-1)^{2-1}=2\times(-1)^1=2\times(-1)=-2 \\ a_3=3\times(-1)^{3-1}=3\times(-1)^2=3\times1=3 \\ \ldots \end{gathered}[/tex]Therefore, an explicit formula for the n-th term of the given sequence, is:
[tex]a_n=n\times(-1)^{n-1}[/tex]