Respuesta :
Answer:
(1)We get that all terms have same [tex](r)[/tex] then It is a Geometric Sequence.
(2)We get Common ratio of given sequence is not same. It is not an geometric sequence.
(3)We get the common ratio is same then it is a Geometric Sequence
(4)We get the common ratio is same then it is a Geometric Sequence.
(5)We get Common ratio of given sequence is not same. It is not an geometric sequence.
Step-by-step explanation:
Here, The Geometric Progression in the form:
[tex]G.P: a, ar, ar^{2}, ar^{3}, ar^{4}, ar^{5}........................................., ar^{n-1}, ar^{n}.[/tex]
Where [tex]a - First\ term\\r - Common \ ratio\\n - Number\ of\ terms\ of\ an\ progression[/tex]
So, Check all that apply.
(1) [tex]4,2,1,\frac{1}{2},\frac{1}{4}[/tex]
For the geometric Sequence Common ratio [tex](r)[/tex] must be same.
⇒ [tex]r= \frac{ar^{n} }{ar^{n-1} }[/tex]
Then, Finding [tex](r)[/tex] for given sequence
[tex]r=\frac{a_{2} }{a_{1} } =\frac{a_{3} }{a_{2} } =\frac{a_{4} }{a_{3} }............................\frac{ar^{n} }{ar^{n-1} } .[/tex]
∴ [tex]r= \frac{2}{4}=\frac{1}{2}=\frac{\frac{1}{2} }{1} =\frac{\frac{1}{4} }{\frac{1}{2} }[/tex]
⇒ [tex]r= \frac{1}{2}=\frac{1}{2}=\frac{1}{2} } = \frac{1}{2} }[/tex]
Clearly,
We get that all terms have same [tex](r)[/tex] then It is a Geometric Sequence.
(2) [tex]-2, 3,-4,5,-6.........[/tex]
Same as above we will check the common ratio
⇒ [tex]r=\frac{3}{-2} =\frac{-4}{3} =\frac{5}{-4}=\frac{-6}{5}[/tex]
⇒ [tex]r=\frac{3}{-2} \neq \frac{-4}{3} \neq \frac{5}{-4}\neq \frac{-6}{5}[/tex]
Clearly,
We get Common ratio of given sequence is not same. It is not an geometric sequence.
(3) [tex]2,6,18,54,162.................[/tex]
Now checking common Ratio [tex](r)[/tex]
⇒ [tex]r=\frac{6}{2} =\frac{18}{6} =\frac{54}{18}=\frac{162}{54}[/tex]
⇒ [tex]r=3=3=3=3[/tex]
Therefore,
We get the common ratio is same then it is a Geometric Sequence.
(4) [tex]-4,-16,-64,-256.........[/tex]
Now checking common Ratio [tex](r)[/tex]
⇒ [tex]r=\frac{-16}{-4} =\frac{-64}{-16} =\frac{-256}{-64}[/tex]
⇒ [tex]r=4=4=4[/tex]
Therefore,
We get the common ratio is same then it is a Geometric Sequence.
(5) [tex]-2,-4,-12,-48,-240............................[/tex]
Now checking common Ratio [tex](r)[/tex]
⇒ [tex]r=\frac{-4}{-2} =\frac{-12}{-4} =\frac{-48}{-12}=\frac{-240}{-48}[/tex]
⇒ [tex]r= 2\neq 3\neq 4\neq 5[/tex]
Clearly,
We get Common ratio of given sequence is not same. It is not an geometric sequence.
