Respuesta :
Answer:
The 9th term for given sequence is 16.777
Therefore the 9th term is [tex]a_{9}=16.777[/tex].
Step-by-step explanation:
Given first three terms of a sequence are 100,80,64,...
Given [tex]a_{1}=100[/tex] ,[tex]a_{2}=80[/tex] , [tex]a_{3}=64[/tex],...
Given sequence is of the form of Geometric sequence
Therefore it can be written as [tex]{\{a,ar,ar^2,...}\}[/tex]
therefore a=100 , ar=80 , [tex]ar^2=64[/tex] ,...
To find common ratio
[tex]r=\frac{a_{2}}{a_{1}}[/tex]
[tex]r=\frac{80}{100}[/tex]
[tex]r=\frac{4}{5}[/tex]
[tex]r=\frac{a_{3}}{a_{2}}[/tex]
[tex]r=\frac{64}{80}[/tex]
[tex]r=\frac{4}{5}[/tex]
Therefore [tex]r=\frac{4}{5}[/tex]
The nth term of the geometric sequence is
[tex]a_{n}=ar^{n-1}[/tex]
To find the 9th tem for the given geometric sequence is
[tex]a_{n}=ar^{n-1}[/tex]
put n=9, a=100 and [tex]r=\frac{4}{5}[/tex]
[tex]a_{9}=100(\frac{4}{5})^{9-1}[/tex]
[tex]=100(\frac{4}{5})^{8}[/tex]
[tex]=100(\frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5})[/tex]
[tex]=100(\frac{256\times 256}{625\times 625})[/tex]
[tex]=100(\frac{65536}{390625})[/tex]
[tex]=100(0.16777})[/tex]
[tex]=16.777[/tex]
Therefore [tex]a_{9}=16.777[/tex]
The 9th term is 16.777
