Question

Which sequences are geometric sequences? Check all that apply. 4, 2, 1, One-half,One-fourth,... −2, 3, −4, 5, −6, … 2, 6, 18, 54, 162, … −4, −16, −64, −256, … −2, −4, −12, −48, −240, …

Answer 1

Answer:

(1)We get that all terms have same $(r)$ 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:

             $G.P: a, ar, ar^{2}, ar^{3}, ar^{4}, ar^{5}........................................., ar^{n-1}, ar^{n}.$

         Where          $a - First\ term\\r - Common \ ratio\\n - Number\ of\ terms\ of\ an\ progression$

So, Check all that apply.

(1)    $4,2,1,\frac{1}{2},\frac{1}{4}$

For the geometric Sequence Common ratio $(r)$ must be same.

⇒       $r= \frac{ar^{n} }{ar^{n-1} }$

Then,     Finding $(r)$ for given sequence

  $r=\frac{a_{2} }{a_{1} } =\frac{a_{3} }{a_{2} } =\frac{a_{4} }{a_{3} }............................\frac{ar^{n} }{ar^{n-1} } .$

   ∴         $r= \frac{2}{4}=\frac{1}{2}=\frac{\frac{1}{2} }{1} =\frac{\frac{1}{4} }{\frac{1}{2} }$

   ⇒       $r= \frac{1}{2}=\frac{1}{2}=\frac{1}{2} } = \frac{1}{2} }$

Clearly,

We get that all terms have same $(r)$ then It is a Geometric Sequence.

(2)  $-2, 3,-4,5,-6.........$

Same as above we will check the common ratio

⇒        $r=\frac{3}{-2} =\frac{-4}{3} =\frac{5}{-4}=\frac{-6}{5}$

⇒        $r=\frac{3}{-2} \neq \frac{-4}{3} \neq \frac{5}{-4}\neq \frac{-6}{5}$

Clearly,

We get Common ratio of given sequence is not same. It is not an geometric sequence.

(3)      $2,6,18,54,162.................$

Now checking common Ratio $(r)$

⇒      $r=\frac{6}{2} =\frac{18}{6} =\frac{54}{18}=\frac{162}{54}$

⇒      $r=3=3=3=3$

Therefore,

We get the common ratio is same then it is a Geometric Sequence.

(4)     $-4,-16,-64,-256.........$

Now checking common Ratio $(r)$

⇒      $r=\frac{-16}{-4} =\frac{-64}{-16} =\frac{-256}{-64}$

⇒      $r=4=4=4$

Therefore,

We get the common ratio is same then it is a Geometric Sequence.

(5) $-2,-4,-12,-48,-240............................$

Now checking common Ratio $(r)$

⇒    $r=\frac{-4}{-2} =\frac{-12}{-4} =\frac{-48}{-12}=\frac{-240}{-48}$

⇒    $r= 2\neq 3\neq 4\neq 5$

Clearly,

We get Common ratio of given sequence is not same. It is not an geometric sequence.

Answer 2

Answer:

1,3,4      Step-by-step explanation: