Question

Find the next three terms in the geometric sequence. 20/63 -10/21 5/7

Answer 1

By definition, a Geometric sequence is that sequence in which a term is found by multiplying the previous one by the "Common ratio". The Common ratio is constant.

In this case you have the folowing Geometric sequence:

$\frac{20}{63},-\frac{10}{21},\frac{5}{7},\ldots$

In order to find the Common ratio of this sequence, you can divide one of the given term by the previous term, as you can see below:

$r=-\frac{10}{21}\div\frac{20}{63}=\frac{(-10)(63)}{(21)(20)}=-\frac{3}{2}$

Therefore, you can set up the following equation for this sequence:

$a_n=(\frac{20}{63})(-\frac{3}{2})^{(n-1)}$

Because the formula for a Geometric sequence is:

$a_n=a_1(r)^{(n-1)}_{}$

Where:

- The nth term is

$a_n$

- The first term is

$a_1$

- The common ratio is "r".

- The term position is "n".

Since, in this case

$\begin{gathered} r=-\frac{3}{2} \\ \\ a_1=\frac{20}{63} \end{gathered}$

You can find the next three terms in the given sequence as following:

$\begin{gathered} a_4=(\frac{20}{63})(-\frac{3}{2})^{(4-1)}=-\frac{15}{14} \\ \\ a_5=(\frac{20}{63})(-\frac{3}{2})^{(5-1)}=\frac{45}{28} \\ \\ a_6=(\frac{20}{63})(-\frac{3}{2})^{(6-1)}=-\frac{135}{56} \end{gathered}$

Therefore, the answer is:

$-\frac{15}{14},\frac{45}{28},-\frac{135}{56}$