Question

Given two terms in a geometric sequence, find the common ratio. Please helpp

Answer 1

In a geometric sequence, consecutive terms have a common ratio:

$\dfrac{a_{n+1}}{a_n}=r\quad \forall n\geq 1$

This means that, in order to build a gometric sequence, we must choose an initial value $a_1$ and a common ratio $r$, and we'll multiply each term by $r$ to get the next one:

$a_1=a_1$

$a_2=ra_1$

$a_3=r^2a_1$

$a_4=r^3a_1$

$a_5=r^4a_1$

This implies that

$\dfrac{a_5}{a_2}=r^3$

And so in this case we have

$\dfrac{10}{80}=r^3 \iff r^3 = \dfrac{1}{8} \iff r=\dfrac{1}{2}$