Question says to use inductive reasoning to predict the next term in the given sequence. -2/3, 1/3, -1/6, 1/12.....
Given sequence can be in AP or GP or any other sequece.
Let's test for GP (Geometric progression)
divide 2nd term by first
[tex]\frac{\left(\frac{1}{3}\right)}{\left(-\frac{2}{3}\right)}=\left(\frac{1}{3}\right)\cdot\left(-\frac{3}{2}\right)=-\frac{1}{2}[/tex]
similarly divide consecutive terms
[tex]\frac{\left(\frac{-1}{6}\right)}{\left(\frac{1}{3}\right)}=\left(\frac{-1}{6}\right)\cdot\frac{3}{1}=-\frac{1}{2}[/tex]
[tex]\frac{\left(\frac{1}{12}\right)}{\left(-\frac{1}{6}\right)}=-\frac{1}{2}[/tex]
all the time we are getting same common ratio -1/2 so that means given sequence is in GP.
Hence we just need to multiply last term by -1/2 to get the answer:
[tex]\left(\frac{1}{12}\right)\cdot\left(-\frac{1}{2}\right)=-\frac{1}{24}[/tex]
Hence final answer is -1/24.
