Question

The 4th and 8th term of a G.P. are 24 and 8/27 respectively. find the 1st term and common ratio

Answer 1

see explanation

Explanation:

The n th term of a geometric progression is

•
`a_(n)` = a₁
`r^(n-1)`

where a₁ is the first term and r the common ratio

given a₄ = 24, then

a₁
`r^(3)` = 24 → (1)

Given a₈ =
`(8)/(27)`, then

a₁
`r^(7)` =
`(8)/(27)` → (2)

Divide (2) by (1)

`r^(4)` =
`((8)/(27) )/(24)` =
`(1)/(81)`

Hence r =
`\sqrt[4]{(1)/(81) }` =
`(1)/(3)`

Substitute this value into (1)

a₁ × (
`(1)/(3)` )³ = 24

a₁ ×
`(1)/(27)` = 24, hence

a₁ = 24 × 27 = 648