Answer:
[tex]\frac{2}{3}[/tex]
Step-by-step explanation:
The sum of an infinite geometric series is expressed according to the formula;
[tex]S_\infty = \dfrac{a}{1-r}[/tex] where;
a is the first term of the series
r is the common ratio
If the sum of an infinite geometric series is three times the first term, this is expressed as [tex]S_\infty = 3a[/tex]
Substitute [tex]S_\infty = 3a[/tex] into the formula above to get the common ratio r;
[tex]3a = \dfrac{a}{1-r} \\\\[/tex]
[tex]cross \ multiply\\\\3a(1-r) = a\\\\3(1-r) = 1\\[/tex]
open the parenthesis
[tex]3 - 3r = 1\\\\[/tex]
subtract 3 from both sides
[tex]3 - 3r -3= 1-3\\\\-3r = -2\\\\r = \frac{2}{3}[/tex]
Hence the common ratio of this series is [tex]\frac{2}{3}[/tex]
