Question

The first three terms of a sequence are given. Round to the nearest thousandth (if necessary).

6, 8, 32/3...
Find the 6th term.

Answer 1

Explanation:

it is a geometric sequence.

the common ratio or factor is 4/3.

I got this by simply trying to get from the 2nd to the 3rd term.

8×x = 32/3

24x = 32

x = 32/24 = 16/12 = 8/6 = 4/3

s1 = 6

s2 = s1 × 4/3 = 6 × 4/3 = 24/3 = 8

s3 = s2 × 4/3 = s1 × 4/3 × 4/3 = 6 × 16/9 = 2 × 16/3 = 32/3

sn = s1 × (4/3)^(n-1)

s6 = 6 × (4/3)⁵ = 6 × 1024/243 = 2 × 1024/81 = 2048/81 =

= 25.284

Answer 2

a₆ ≈ 25.284

Explanation:

There is a common ratio between consecutive terms , that is

8 ÷ 6 =
`(32)/(3)` ÷ 8 =
`(4)/(3)`

This indicates the sequence is geometric with nth term

`a_(n)` = a₁
`r^(n-1)`

where a₁ is the first term and r the common ratio

Here a₁ = 6 and r =
`(4)/(3)` , then

a₆ = 6 ×
`((4)/(3)) ^(5)` = 6 ×
`(1024)/(243)` =
`(6(1024))/(243)` ≈ 25.284 ( to the nearest thousandth )