Exercises

$\frac{1}{3}$

2. Given a point A for the interval $[0,\;1)$ of a straight line one can construct the following sequence of rational numbers.

A₀ = 0

$A_1 = \frac{a_1}{8}$

$A_2 = \frac{a_1}{8} + \frac{a_2}{8^2}$

$\vdots$

$A_n = \frac{a_1}{8} + \frac{a_2}{8^2}+ \dots + \frac{a_n}{8^n}$

where each of the coefficients {a_i} can take only the following values:

$0,\;1,\;2,\;3,\;4,\;5,\;6,\;7$

For example, a_{n + 1} = 5 if

$\frac{5}{8} \le A - (\frac{a_1}{8} + \frac{a_2}{8^2}+ \dots + \frac{a_n}{8^n}) < \frac{6}{8}$

The corresponding representation

$0. a_1 a_2 \dots$

is called octal representation of A.

Write octal representation for

$\frac{3}{5} .$

$\sqrt{n^2 -n} -n \;\;\;$

$n \;\;\;$

$(-1)^n \;\;\;$

$\frac{n-1}{2n+1} \;\;\;$

where $n=1,\;2,\;\dots$