2010006910
Level:
B
Express the repeating decimal \( 0.4\overline{32} \) as a fraction in lowest terms.
\( \frac{214}{495} \)
\( \frac{98}{225} \)
\( \frac{16}{495} \)
\( \frac{8}{225} \)
Infinite series
2010006909
Level:
A
Evaluate the following infinite sum.
\[
-\frac{3}
{4} + \frac{1}
{4} -\frac{3}
{8} + \frac{1}
{8} - \frac{3}
{16} +\cdots
\]
\( -1 \)
\(-\frac{1}
{4}\)
\(-\frac{1}
{2}\)
\( -2 \)
2010006908
Level:
A
Find the sum of the following infinite series.
\[
\sum _{n=1}^{\infty }\left (\frac{\sqrt{5} - 1}
{\sqrt{5}} \right )^{n-1}
\]
\( \sqrt{5} \)
\( \frac{\sqrt{5}}{5}\)
\( \frac{\sqrt{5}-1}{5}\)
Series diverges.
2010006907
Level:
B
Find all the values of \(x\)
such that the following infinite series is convergent.
\[
1 + 2x - 3 + (2x-3)^{2} + (2x - 3)^{3}+\cdots
\]
\(x\in (1;2)\)
\(x\in (-\infty ;-1)\)
\(x\in (1;+\infty )\)
\(x\in \mathbb{R}\)
2010006906
Level:
B
Solve the following equation.
\[
1 + 3x + 9x^{2} + \cdots = 2
\]
\(x = \frac{1}
{6}\)
\(x = -\frac{1}
{3}\)
\(x = \frac{1}
{3}\)
The equation has no solution.
2010006905
Level:
A
Evaluate the following infinite sum.
\[
\sum _{n=1}^{\infty }\left (-\frac{3}
{5}\right )^{n}
\]
\(- \frac{3}
{8}\)
\(- \frac{3}
{2}\)
\(\frac{3}
{2}\)
\(\frac{3}
{8}\)
2010006904
Level:
B
We are given the equation
\[ \sum\limits_{n=0}^{\infty}\frac{(x+2)^n}{3^n}=\frac{x+3}{2x+1} \]
with the unknown \( x \) being a real number. What is the set of all its solutions?
\( \{ 0\} \)
\( \{ -8;0 \} \)
\( \{ \} \)
\( \{ -6;2 \} \)
\( \{ -8 \} \)
2010006903
Level:
B
What is the sum of the repeating numbers \( 0.\overline{45} \) and \( 0.2\overline{81} \)?
\( \frac{81}{110} \)
\( \frac{59}{110} \)
\( \frac{110}{81} \)
\( \frac{36}{110} \)
\( \frac{81}{100} \)
2010006902
Level:
A
The sum of the infinite geometric series
\[ \left(\sqrt3-1\right)+\left(\sqrt3-1\right)^2+\left(\sqrt3-1\right)^3+\dots \]
is equal to:
\( 1+\sqrt3 \)
\(\sqrt3-1 \)
\( \frac{3-\sqrt{3}}{3}\)
\( \frac{\sqrt{3}-3}{3}\)
\( \sqrt3\)
2010006901
Level:
A
We are given an infinite geometric series:
\[ \left(\sqrt2-\sqrt7\right)+\left(2-\sqrt{14}\right)+\left(2\sqrt2-2\sqrt{7}\right)+\dots\text{ .} \]
What is the value of its common ratio?
\( \sqrt{2} \)
\( \sqrt{2} - \sqrt{7}\)
\(2\)
\(7\)
\( \sqrt{2} - 2\)