In a geometric sequence, consecutive terms have a common ratio:
[tex]\dfrac{a_{n+1}}{a_n}=r\quad \forall n\geq 1[/tex]
This means that, in order to build a gometric sequence, we must choose an initial value [tex]a_1[/tex] and a common ratio [tex]r[/tex], and we'll multiply each term by [tex]r[/tex] to get the next one:
[tex]a_1=a_1[/tex]
[tex]a_2=ra_1[/tex]
[tex]a_3=r^2a_1[/tex]
[tex]a_4=r^3a_1[/tex]
[tex]a_5=r^4a_1[/tex]
This implies that
[tex]\dfrac{a_5}{a_2}=r^3[/tex]
And so in this case we have
[tex]\dfrac{10}{80}=r^3 \iff r^3 = \dfrac{1}{8} \iff r=\dfrac{1}{2}[/tex]
