Question

Which of the following is a geometric sequence?

Answer 1

Answer:

Option C. is the geometric sequence.

Step-by-step explanation:

In a geometric sequence there should be a common ratio in each successive term.

Now we will check the common ratio in each option.

A). $1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}........$

In this sequence if we find the common ratio

Ratio of second and first term = $\frac{\text{Second term}}{\text{First term}}$

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

Similarly ratio of third and second term = $\frac{\text{Third term}}{\text{Second term}}$

= $\frac{\frac{1}{6}}{\frac{1}{2}}$

= $\frac{2}{6}$

= $\frac{1}{3}$

Here ratios are not common. Therefore, sequence is not a geometric sequence.

B). -1, -1, 1, -1, -1..............

ratio of second and first term = $\frac{-1}{-1}=1$

ratio of third and second term = $\frac{1}{-1}=-1$

Ratios are not equal so sequence is not a geometric sequence.

C). 1, 3, 9, 27...............

ratio of second and first term = $\frac{3}{1}=3$

ratio of third and second term = $\frac{9}{3}=3$

Both the ratios are same therefore, sequence is geometric.

D). 3, 6, 9, 12..............

ratio of second and first term = $\frac{6}{3}=2$

ratio of third and second term = $\frac{9}{6}=\frac{3}{2}$

Ratios are not equal so sequence is not a geometric sequence.

Answer 2

Answer:

Option C. $1,3,9,27, ...$ is a geometric sequence in which the common ratio is 3.

Step-by-step explanation:

In geometric sequence, the the ratio between two consecutive numbers is the same.

Now, consider the first case. The given sequence is $1, \dfrac{1}{2},\dfrac{1}{6}, \dfrac{1}{24}, ...$.

In the above sequence, The ratios between consecutive numbers are:

$\dfrac{1}{2}:1=\dfrac{1}{2}\\\dfrac{1}{6}:\dfrac{1}{2}=\dfrac{1}{3}$

So, the ratio between two consecutive numbers is not the same and hence, it is not a geometric sequence.

Now, consider the second case. The given sequence is $-1,-1,1,-1,-1,1, ...$.

In the above sequence, The ratios between consecutive numbers are:

$-1:-1=1\\1:-1=-1$

So, the ratio between two consecutive numbers is not the same and hence, it is not a geometric sequence.

Now, consider the third case. The given sequence is $1,3,9,27, ...$.

In the above sequence, The ratios between consecutive numbers are:

$3:1=3\\9:3=3\\27:9=3$

So, the ratio between two consecutive numbers is the same and hence, it is  a geometric sequence.

Now, consider the fourth case. The given sequence is $3,6,9,12,...$.

In the above sequence, The ratios between consecutive numbers are:

$6:3=2\\9:6=\dfrac{3}{2}$

So, the ratio between two consecutive numbers is not the same and hence, it is not a geometric sequence.

Therefore, option C. $1,3,9,27, ...$ is a geometric sequence in which the common ratio is 3.

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https://brainly.com/question/13008517?referrer=searchResults