Question

Which sequence is geometric and has `1/4` as its fifth term and `1/2` as the common ratio?

Answer 1

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

Explanation:

Fifth term (a₅) =
`(1)/(4)`

Common ratio (r) =
`(1)/(2)`

n = 5

The formula to find the
`n^(th)` term of a geometric sequence is

a₅ = a₁r⁽ⁿ⁻¹⁾

=>
`(1)/(4)` = a₁*
`((1)/(2))^((5-1))`

=>
`(1)/(4)` = a₁*
`((1)/(2))^((4))`

=>
`(1)/(4)` = a₁*
`((1)/(2))*((1)/(2))*((1)/(2))*((1)/(2))`

=>
`(1)/(4)` = a₁*
`(1*1*1*1)/(2*2*2*2)`

=>
`(1)/(4)` = a₁*
`((1)/(16))`

Flip the sides of the equation

a₁*
`((1)/(16))` =
`(1)/(4)`

Multiply both sides by 16

a₁*
`((1)/(16))`*16 =
`(1)/(4)`*16

Cancelling out the 16's from the top and bottom of the left side

a₁ =
`(16)/(4)`

=> a₁ = 4

So, first term of the geometric sequence is 4.

Common ratio =
`(1)/(2)`

Second term = First term * Common ratio

= 4*
`(1)/(2)`

=
`(4)/(2)`

= 2

Third term = Second Term * Common ratio

= 2 *
`(1)/(2)`

=
`(2)/(2)`

= 1

Fourth term = Third Term * Common ratio

= 1 *
`(1)/(2)`

=
`(1)/(2)`

Fifth term (given) =
`(1)/(4)`

So, the geometric sequence would be

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