Last one I'm stuck on:
$\displaystyle a_n$ = ?
{ $\displaystyle 1, -\frac{2}{3}, .4444444444444444, - .296296296296296296,.....$}
Here is the explanation about third and 4th term
$\displaystyle x = .4444444444444444.. ---> (1)$
$\displaystyle 10x = 4.44444...----->(2)$
(2) - (1)
$\displaystyle 9x= 4 $
$\displaystyle
x = \frac{4}{9} = \frac{ 2\times 2}{ 3 \times 3} $
$\displaystyle x = -0.296296296296296296 ..---> (1) $
$\displaystyle 1000x =- 296.296296......----> (2)$
(2)-(1)
$\displaystyle 999x = -296 $
$\displaystyle x = \frac{-296}{999} = \frac{4 \times \not 37 \times -2}{9 \times \not 37 \times 3 }$