Answer
a) We note that the given decimal is a recurring decimal. Thus, it is probable that $x \lt 1$.
b) The sum of the series is $x=1$.
c) We know that the number one has two decimal representations.
d) All rational numbers whose decimal representation is non-recurring have more than one decimal representation. For example, 0.4 can be written as 0.3999... and as 0.4000...
Work Step by Step
a) We note that the given decimal is a recurring decimal. Thus, it is probable that $x \lt 1$.
b) Let us start by representing $x$ as the sum of a series.
$x=0.9+0.09+0.009+0.0009+...$
$=9(0.1+0.01+0.001+0.0001)$
First term $a= 0.1$ and $r=\frac{1}{10}$
Sum of the series is
$x=9(\frac{0.1}{1-0.1})$
$=\frac{0.9}{0.9}$
$=1$
c) We know that the number one has two decimal representations (1.000... and 0.999...).
d) All rational numbers whose decimal representation is non-recurring have more than one decimal representation. For example, 0.4 can be written as 0.3999... and as 0.4000...
