Question

Determine the common ratio of a geometric sequence if the second term is -4 and the fifth term is -27/2.A- -3/2B- 3/2C- -27/16D- 27/16

Answer 1

A geometric sequence is a sequence in which each term is found by multiplying the preceding term by the same value. Its general term is

$a_n=a_1r^{n-1}$

The value r is called the common ratio.

On the sequence of our problem, we know the second and fifth term. If we substitute those values on the general term presented, we have

$\begin{gathered} a_2=a_1r^{2-1}=a_1r=-4 \\ a_5=a_1r^{5-1}=a_1r^4=-\frac{27}{2} \end{gathered}$

If we divide the fifth term by the second term, we're going to have

$\frac{a_5}{a_2}=\frac{a_1r^4}{a_1r}=r^3\operatorname{\implies}r^3=\frac{(-27\/2)}{(-4)}$

Solving for r, we have

$\begin{gathered} r^3=\frac{(-27\/2)}{(-4)} \\ r^3=\frac{27}{8} \\ r=\sqrt[3]{\frac{27}{8}} \\ r=\frac{3}{2} \end{gathered}$

The common ratio is 3/2.