1003108802
Level:
B
Express the repeating decimal \( 0.5\overline{83} \) as a fraction in lowest terms.
\( \frac{289}{495} \)
\( \frac{83}{990} \)
\( \frac{495}{289} \)
\( \frac{298}{495} \)
\( \frac{583}{1\,000} \)
Infinite series
1003108801
Level:
B
Express the repeating decimal \( 2.\overline{4} \) as a fraction in lowest terms.
\( \frac{22}9 \)
\( \frac49 \)
\( \frac{22}7 \)
\( \frac{12}5 \)
\( \frac9{22} \)
1003108713
Level:
A
The value of the expression
\[ \sum\limits_{n=1}^{\infty}\left(\frac13\right)^n \]
is equal to:
\( \frac12 \)
\( 2 \)
\( 1 \)
\( \frac14 \)
\( \frac23 \)
1003108712
Level:
A
The value of the expression
\[ \sum\limits_{n=1}^{\infty}(-1)^n\cdot\left(\frac23\right)^n \]
is equal to:
\( -\frac25 \)
\( \frac25 \)
\( -\frac23 \)
\( \frac23 \)
\( \frac52 \)
1003108711
Level:
A
The sum of the infinite geometric series
\[ 1-\frac34+\frac9{16}-\frac{27}{64}+\dots \]
is equal to:
\( \frac47 \)
\( \frac14 \)
\( 4 \)
\( \frac74 \)
\( \frac57 \)
1003108710
Level:
A
The sum of the infinite geometric series
\[ \left(\sqrt5-2\right)+\left(\sqrt5-2\right)^2+\left(\sqrt5-2\right)^3+\dots \]
is equal to:
\( \frac{\sqrt5-1}4 \)
\( \frac{\sqrt5}4 \)
\( \frac{\sqrt5+1}4 \)
\( \frac{\sqrt5+3}4 \)
\( \frac{\sqrt5-1}2 \)
1003108709
Level:
A
We are given an infinite geometric series:
\[ \sum\limits_{n=1}^{\infty} \frac{\sqrt3}{2^{n-1}}\text{ .} \]
What is the value of its first term \( a_1 \)?
\( \sqrt3 \)
\( \frac{\sqrt3}2 \)
\( \frac{\sqrt3}4 \)
\( \frac12 \)
\( 3 \)
1003108708
Level:
A
We are given an infinite geometric series:
\[ \sum\limits_{n=1}^{\infty} \left(-\frac12\right)^{n-1} \text{ .} \]
What is the value of its second term \( a_2 \)?
\( -\frac12 \)
\( 1 \)
\( \frac12 \)
\( \frac14 \)
\( -1 \)
1003108707
Level:
A
We are given an infinite geometric series:
\[ \left(\sqrt5-\sqrt3\right)+\left(5-\sqrt{15}\right)+\left(5\sqrt5-5\sqrt{3}\right)+\dots\text{ .} \]
What is the value of its common ratio?
\( \sqrt5 \)
\( \sqrt5-\sqrt3 \)
\( \sqrt5-\sqrt3+5 \)
\( \sqrt5+5 \)
\( 5 \)
1003108706
Level:
A
We are given an infinite geometric series:
\[ \frac23-\frac49+\frac8{27}-\frac{16}{81}+\dots\text{ .} \]
What is the value of its common ratio?
\( -\frac23 \)
\( \frac23 \)
\( \frac29 \)
\( -\frac29 \)
\( \frac4{27} \)