Answer:
The next fraction in the given geometric sequence is [tex]\frac{40}{243}[/tex]
Therefore [tex]a_5=\frac{40}{243}[/tex]
Step-by-step explanation:
Given geometric sequence is
[tex]\frac{5}{6}, \frac{5}{9}, \frac{10}{27}, \frac{20}{81}, \ldots[/tex]
To find the 5th term of the given geometric sequence:
Let [tex]a_1=\frac{5}{6}, a_2=\frac{5}{9}, a_3=\frac{10}{27}, a_4=\frac{20}{81}[/tex] etc.
First find the common ratio
[tex]r=\frac{a_2}{a_1}[/tex]
[tex]=\frac{\frac{5}{9}}{\frac{5}{6}}[/tex]
[tex]=\frac{5}{9}\times \frac{6}{5}[/tex]
Therefore [tex]r=\frac{2}{3}[/tex]
[tex]r=\frac{a_3}{a_2}[/tex]
[tex]=\frac{\frac{10}{27}}{\frac{5}{9}}[/tex]
[tex]=\frac{10}{27}\times \frac{9}{5}[/tex]
Therefore [tex]r=\frac{2}{3}[/tex]
Therefore the common ratio [tex]r=\frac{2}{3}[/tex]
The nth term of geometric sequence is [tex]a_n=a_1r^{n-1}[/tex]
Put [tex]n=5, a_1=\frac{5}{6}[/tex] and [tex]r=\frac{2}{3}[/tex] in the above equation we get
[tex]a_5=\frac{5}{6}(\frac{2}{3})^{5-1}[/tex]
[tex]=\frac{5}{6}(\frac{2}{3})^4[/tex]
[tex]=\frac{5}{6}(\frac{2}{3}\times \frac{2}{3}\times \frac{2}{3}\times \frac{2}{3})[/tex]
[tex]=\frac{5}{3}(\frac{8}{81})[/tex]
[tex]a_5=\frac{40}{243}[/tex]
Therefore [tex]a_5=\frac{40}{243}[/tex]
Therefore the next fraction in the given geometric sequence is [tex]\frac{40}{243}[/tex]
