Question

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Answer 1

Answer:c

15th term of the geometric sequence is 0.0122

Explanation:

We are given that,

The fourth term and ninth term of the geometric sequence are
`-25` and
`(25)/(32)`.

Since, the general expression for the nth term of a geometric sequence is
`a_(n)=ar^(n-1)`.

So, we get,

`a_(4)=ar^(4-1)=ar^(3)=-25` and
`a_(9)=ar^(9-1)=ar^(8)=(25)/(32)`

Dividing both the terms, we get,

`(a_(4))/(a_(9))=(-25)/((25)/(32))`

i.e.
`(ar^(3))/(ar^(8))=(-25)/((25)/(32))`

i.e.
`(1)/(r^(5))=-32`

i.e.
`r^5=(-1)/(32)`

i.e.
`r^5=-0.03125`

i.e. r= -0.5

So, we get,

`a_(4)=ar^(3)=-25`

i.e.
`a(-0.5)^(3)=-25`

i.e.
`a=(-25)/(-0.125)`

i.e. a= 200.

Then, we have,

`a_(15)=ar^(15-1)`

i.e.
`a_(15)=ar^(14)`

i.e.
`a_(15)=200* (-0.5)^(14)`

i.e.
`a_(15)=200* 6.1* 0.00001`

i.e.
`a_(15)=1220* 0.00001`

i.e.
`a_(15)=0.122`

Thus, the 15th term of the geometric sequence is 0.0122

Answer 2

15th term will be
`(25)/(2048) }`

Explanation:

In a geometric progression we know any term of the series is represented as
`T(n) = a(r)^(n-1)`

where a = first term, r = common ratio, and n = number of term

Now as per question 4th term is -25 then as pr formula

`T(4) = a (r)^(4-1) = ar^(3) = -25`------(1)

Now the ninth term is 25/32

`T(9) = ar^(9-1) = ar^(8) = (25)/(32)`-----(2)

Now we put the value of a from equation 1 to equation 2

`a =(-25)/(r^(3) )`

`(-25)/(r^(3) ) (r^(8))=(25)/(32)\\-25r^(5) = (25)/(32\\) or -r^(5)=(1)/(32)`

`r^(5) = -(1)/(32)`

`r = -\frac{1}{\sqrt[5]{32}}  = -(1)/(2)`

therefore a =
`a(-(1)/(2))^(8) =(25)/(32)`

`a=(25)/(32)(2)^(8)`

`a = (25)/(2^(5) ) (2^(8)) =25(2)^(3) =200`

Therefore 15th term will be

`200(-(1)/(2)) ^(15-1) = 200((1)/(2))^(14)= (25)/(2^(11) )= (25)/(2048)`