Answer:
For the sequence is [tex]-\frac{2}{3}[/tex] ,-4 ,-24 ,-144 ,...
Hence the formula [tex]f(x)=-\frac{2}{3}(6)^{x-1}[/tex] for x=1,2,3,... represents the given geometric sequence
Step-by-step explanation:
Given sequence is [tex]-\frac{2}{3}[/tex] ,-4 ,-24 ,-144 ,...
To find the formula to describe the given sequence :
Let [tex]a_1=\frac{-2}{3}[/tex] ,[tex]a_2=-4[/tex] ,[tex]a_3=-24[/tex],...
First find the common ratio
[tex]r=\frac{a_2}{a_1}[/tex] here [tex]a_1=\frac{-2}{3}[/tex] and,[tex]a_2=-4[/tex]
[tex]=\frac{-4}{\frac{-2}{3}}[/tex]
[tex]=\frac{4\times 3}{2}[/tex]
[tex]=\frac{12}{2}[/tex]
[tex]r=6[/tex]
[tex]r=\frac{a_3}{a_2}[/tex] here [tex]a_2=-4[/tex] and [tex]a_3=-24[/tex]
[tex]=\frac{-24}{-4}[/tex]
[tex]=6[/tex]
[tex]r=6[/tex]
Therefore the common ratio is 6
Therefore the given sequence is geometric sequence
The nth term of the geometric sequence is
[tex]a_n=a_1r^{n-1}[/tex]
The formula which describes the given geometric sequence is
[tex]f(x)=a_1r^{x-1}[/tex] for x=1,2,3,...
[tex]=\frac{-2}{3}6^{x-1}[/tex] for x=1,2,3,...
Now verify that [tex]f(x)=a_1r^{x-1}[/tex] for x=1,2,3,... represents the given geometric sequence or not
put x=1 and the value of [tex]a_1[/tex] in [tex]f(x)=a_1r^{x-1}[/tex] for x=1,2,3,...
we get [tex]f(1)=-\frac{2}{3}(6)^{1-1}[/tex]
[tex]=-\frac{2}{3}(6)^0[/tex]
[tex]=-\frac{2}{3}[/tex]
Therefore [tex]f(1)=-\frac{2}{3}[/tex]
put x=2 we get [tex]f(2)=-\frac{2}{3}(6)^{2-1}[/tex]
[tex]=-\frac{2}{3}(6)^1[/tex]
[tex]=-\frac{12}{3}[/tex]
Therefore [tex]f(2)=-4[/tex]
put x=3 we get [tex]f(3)=-\frac{2}{3}(6)^{3-1}[/tex]
[tex]=-\frac{2}{3}(6)^2[/tex]
[tex]=-\frac{2(36)}{3}[/tex]
Therefore [tex]f(3)=-24[/tex]
Therefore the sequence is f(1),f(2),f(3),...
Therefore the sequence is [tex]-\frac{2}{3}[/tex] ,-4 ,-24 ,-144 ,...
Hence the formula [tex]f(x)=a_1r^{x-1}[/tex] for x=1,2,3,... represents the given geometric sequence is verified
Therefore the formula [tex]f(x)=-\frac{2}{3}(6)^{x-1}[/tex] for x=1,2,3,... represents the given geometric sequence
