The formula which can be used to describe the sequence is:
[tex]f(x+1)=2f(x)[/tex]
The sequence is given by:
[tex]-2\dfrac{2}{3},-5\dfrac{1}{3},-10\dfrac{2}{3},-21\dfrac{1}{3},-42\dfrac{2}{3},.....[/tex]
The terms of the sequence could also be written as:
[tex]\dfrac{-8}{3},\dfrac{-16}{3},\dfrac{-32}{3},\dfrac{-64}{3},\dfrac{-128}{3},....[/tex]
This means that the first term of the sequence is:
[tex]f(1)=\dfrac{-8}{3}[/tex]
The second term of the sequence is:
[tex]f(2)=\dfrac{-16}{3}\\\\i.e.\\\\f(2)=2\times \dfrac{-8}{3}\\\\i.e.\\\\f(2)=2f(1)[/tex]
[tex]f(3)=\dfrac{-32}{3}\\\\i.e.\\\\f(3)=2\times \dfrac{-16}{3}\\\\i.e.\\\\f(3)=2f(2)[/tex]
and so on.
Hence, the formula is given by:
[tex]f(x+1)=2f(x)[/tex]
