Question 1126579
Arithmetic Sequence 

definition of nth term: {{{a[n] = a[1] + d (n-1)}}}

{{{3}}},....{{{-6}}},....{{{12}}},.....{{{4}}},.....{{{20}}},-----------------{{{213}}}
-- {{{-9}}}.....{{{18}}}.....{{{-8}}}.....{{{16}}}--------------------{{{193}}}
........{{{27}}}.....{{{-26}}}....{{{24}}}----------------------{{{177}}}
........... {{{-53}}}.....{{{50}}}------------------------{{{153}}}
 .............   {{{103}}}--------------------------{{{103}}}

Differences didn't converge, there is {{{no}}}{{{ common}}}{{{ difference}}}; so, it is not Arithmetic Sequence 

{{{3}}},....{{{-6}}},....{{{12}}},.....{{{4}}},.....{{{20}}}

Geometric Sequence 

definition of nth term: {{{ a[n]= a[1]*r^(n-1)}}}

find out if there is common ratio:

{{{a[1]=3}}} and {{{a[2]=-6}}}

{{{a[2]= a[1]* r^(2-1)}}}

{{{a[2]/a[1]=r}}}

{{{r=-6/3}}}

{{{r=-2}}}

check second and third term

{{{a[2]=-6}}} and {{{a[3]=12}}}

{{{a[3]/ a[2]= r^(3-1)}}}

{{{12/ -6= r^2}}}

{{{-2= r^2}}}

{{{r=sqrt(-2)}}}

since not equal, there is {{{no}}}{{{ common}}}{{{ ratio}}} 

since your sequence is neither arithmetic nor geometric sequence, there is no way to find next term