Question 598796
A geometric sequence is a sequence of numbers (called terms) where each number is the product of the one before times a fixed number, called the common ratio. So, with
{{{a[n]}}}=the nth term
{{{a[n+1]}}}=the term after {{{a[n]}}} and
{{{r}}}=the common ratio
we have the relations
{{{a[n+1]=a[n]*r}}} <--> {{{a[n+1]/a[n]=r}}}
and if we call the first term {{{a[1]}}} then
{{{a[n]=a[1]*r^(n-1)}}}
 
In 4, 12, 36,108... , {{{a[1]=4}}}, and we see that {{{r=12/4=3}}}, the same as {{{36/12=3}}} and {{{108/36=3}}}
so applying {{{a[n]=a[1]*r^(n-1)}}} we find that
{{{a[10]=4*3^(10-1)=4*3^9=4*19683=78732}}}
 
In 972, -324, 108, -36... , {{{a[1]=972}}}, and we see that {{{r=972/(-324)=-1/3}}},
so applying {{{a[n]=a[1]*r^(n-1)}}} we find that
{{{a[10]=972*(-1/3)^(10-1)=972*(-1/3)^9=-972*(1/3^9)=-972/3^9=-4*3^5/3^9=-4/3^4=-4/81}}}