Question

The second term in a geometric sequence is 20. The fourth term in the same sequence is 11.25. What is the common ratio

Answer 1

Let $a_n$ be the n'th term of the sequence.

so $a_1$ is the first term, $a_2$ the second term and so on...

In a geometric sequence with $a_1=c$, and common ratio r, the terms are as follows:

$a_1=c$
$a_2=cr$
$a_3=c r^{2}$
$a_4=c r^{3}$
.
.
that is, each term is its previous term times the common ratio r.

In our example

$a_2=cr=20$ and $a_4=c r^{3}=11.25$

$\frac{a_2}{a_4}= \frac{cr}{c r^{3}}= \frac{1}{ r^{2}}= \frac{20}{11.25}= 1.778$

so $r^{2} =1/1.778=0.56$

$r= \sqrt{0.56}= 0.75$


Answer: r=0.75