Question

Negative 2 and two-thirds, negative 5 and one-third, negative 10 and two-thirds, negative 21 and one-third, negative 42 and two-thirds, ellipsis Which formula can be used to describe the sequence?

Answer 1

`f(n) = -(8)/(3)n`

Explanation:

Given

`-2(2)/(3), -5(1)/(3), -10(2)/(3), -21(1)/(3), -42(2)/(3)`

Required

The explicit formula

The above sequence is an arithmetic sequence and it is bounded by:

`f(n) = a + (n - 1)d`

Where

`a = -2(2)/(3)` -- the first term

`d = f(2) -f(1)`

So, we have:

`d = -5(1)/(3) - -2(2)/(3)`

`d = -5(1)/(3) +2(2)/(3)`

Express as improper fraction

`d = -(16)/(3) +(8)/(3)`

Take LCM

`d = (-16+8)/(3)`

`d = -(8)/(3)`

So, we have:

`f(n) = a + (n - 1)d`

`f(n) = -2(2)/(3) + (n - 1) * -(8)/(3)`

Express all fractions as improper

`f(n) = -(8)/(3) + (n - 1) * -(8)/(3)`

Open brackets

`f(n) = -(8)/(3) -(8)/(3)n +(8)/(3)`

Collect like terms

`f(n) = -(8)/(3)n +(8)/(3)-(8)/(3)`

`f(n) = -(8)/(3)n`