Question

Write the nth term of the sequence -1/2,1/3,-2/9,4/27,-8/81?

Answer 1

`a_n=(-1/2)(-2/3)^(n-1), n>=1`

Explanation:

We need to determine whether all of the terms we're given share something in common that can be used to write a formula.

It does not look like an arithmetic sequence, where everything differs by an added or subtracted constant, but geometric.

We can verify this and determine the common ratio each term shares by dividing each term by the preceding term and seeing if the result is always the same:

`(1/3)/(-1/2)=1/3*-2=-2/3`

`(-2/9)/(1/3)=-2/9*3=-6/9=-2/3`

`(4/27)/(-2/9)=4/27*-9/2=-18/27=-2/3`

`(-8/81)/(4/27)=-8/81*27/4=54/81=-2/3`

So, this is indeed a geometric sequence with the common ratio `r=-2/3.`

The standard form of a geometric sequence where `n>=1` is given by

`a_n=a(r)^(n-1)` where `a` is the first term in the sequence, `r` is the common ratio.

Here, `a=-1/2, r=-2/3,` so,

`a_n=(-1/2)(-2/3)^(n-1), n>=1`