To find the common ratio of a geometric sequence, we need to divide any term by its preceding term. In this sequence: [tex]\(625, 125, 25, 5, 1, \ldots \)[/tex], we can take any consecutive terms and calculate this ratio.
Let's take the first two terms of the sequence: 625 and 125.
The common ratio [tex]\(r\)[/tex] is given by:
[tex]\[ r = \frac{\text{second term}}{\text{first term}} = \frac{125}{625} \][/tex]
Now, let's simplify this fraction:
[tex]\[ \frac{125}{625} = \frac{125 \div 125}{625 \div 125} = \frac{1}{5} = 0.2 \][/tex]
So, the common ratio of the given geometric sequence is [tex]\( 0.2 \)[/tex].
