Question

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Answer 1

see explanation

Explanation:

(a)

Using the recursive rule and a₁ = 4 , then

a₂ = 2(
`(1)/(3)` + a₁ ) = 2(
`(1)/(3)` + 4) = 2 × 4
`(1)/(3)` = 2 ×
`(13)/(3)` =
`(26)/(3)`

a₃ = 2(
`(1)/(3)` + a₂) = 2(
`(1)/(3)` +
`(26)/(3)` ) = 2 ×
`(27)/(3)` = 2 × 9 = 18

a₄ = 2(
`(1)/(3)` + a₃) = 2(
`(1)/(3)` + 18) = 2 × 18
`(1)/(3)` = 2 ×
`(55)/(3)` =
`(110)/(3)`

a₅ = 2(
`(1)/(3)` + a₄) = 2(
`(1)/(3)` +
`(110)/(3)`) = 2 ×
`(111)/(3)` = 2 × 37 = 74

------------------------------------------------------------------------

(b)

Using the recursive rule and a₁ = 6 , then

a₂ =
`(2)/(a_(1) )` =
`(2)/(6)` =
`(1)/(3)`

a₃ =
`(3)/(a_(2) )` =
`(3)/((1)/(3) )` = 9

a₄ =
`(4)/(a_(3) )` =
`(4)/(9)`

a₅ =
`(5)/(a_(4) )` =
`(5)/((4)/(9) )` = 5 ×
`(9)/(4)` =
`(45)/(4)`

------------------------------------------------------------------------

(c)

The sequence has a common ratio

r =
`((2)/(3) )/((1)/(6) )` =
`((8)/(3) )/((2)/(3) )` = 4

Multiply the previous terms by 4 , then

a₄ = 4 ×
`(8)/(3)` =
`(32)/(3)`

a₅ = 4 ×
`(32)/(3)` =
`(128)/(3)`