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

1,3,4 Explanation:

Answer 2

(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.

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,(1)/(2),(1)/(4)`

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

⇒
`r= (ar^(n) )/(ar^(n-1) )`

Then, Finding
`(r)` for given sequence

`r=(a_(2) )/(a_(1) ) =(a_(3) )/(a_(2) ) =(a_(4) )/(a_(3) )............................(ar^(n) )/(ar^(n-1) ) .`

∴
`r= (2)/(4)=(1)/(2)=((1)/(2) )/(1) =((1)/(4) )/((1)/(2) )`

⇒
`r= (1)/(2)=(1)/(2)=(1)/(2) } = (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=(3)/(-2) =(-4)/(3) =(5)/(-4)=(-6)/(5)`

⇒
`r=(3)/(-2) \\eq (-4)/(3) \\eq (5)/(-4)\\eq (-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=(6)/(2) =(18)/(6) =(54)/(18)=(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=(-16)/(-4) =(-64)/(-16) =(-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=(-4)/(-2) =(-12)/(-4) =(-48)/(-12)=(-240)/(-48)`

⇒
`r= 2\\eq 3\\eq 4\\eq 5`

Clearly,

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