Question

What is the 6th term of the geometric sequence where a1 = 1,024 and a4 = −16? 1 −0.25 −1 0.25

Answer 1

Option 3rd is correct

`a_6 = -1`

Explanation:

The nth term for the geometric sequence is given by:

`a_n = a_1 \cdot r^(n-1)`

where,

`a_1` is the first term

r is the common ratio

n is the number of terms.

As per the statement:

`a_1 = 1024`

`a_4 = -16`

For n = 4, we have;

`a_4 = a_1 \cdot r^3`

⇒
`-16 = 1024 \cdot r^3`

Divide both sides by 1024 we have;

`-(1)/(64) =r^3`

⇒
`r =\sqrt[3]{-(1)/(64)}=\sqrt[3]{-(1)/(4^3)}`

⇒
`r = -(1)/(4) = -0.25`

We have to find the value of 6th term.

for n = 6

`a_6 = 1024 \cdot (-0.25)^5 = 1024 \cdot (-0.0009765625) = -1`

⇒
`a_6 = -1`

Therefore, the 6th term of the geometric sequence is, -1

Answer 2

The n-th term is given by

`a_n=a_1\cdot r^((n-1))\qquad\text{where r is the common ratio}`

Then we can find the common ratio from the given terms.

`(a_4)/(a_1)=(a_1\cdot r^((4-1)))/(a_1)=r^3=(-16)/(1024)=\left((-1)/(4)\right)^3\\\\r=(-1)/(4)\\\\a_6=1024\left((-1)/(4)\right)^5=-1`

The appropriate choice is -1.