9000063406
Level:
A
Evaluate the following infinite sum.
\[
\sum _{n=1}^{\infty }\left (-\frac{1}
{2}\right )^{n+2}
\]
\(- \frac{1}
{12}\)
\(-\frac{1}
{8}\)
\(\frac{1}
{2}\)
\(1\)
Infinite Series
9000063409
Level:
B
Solve the following equation.
\[
1 + 2x + 4x^{2} + 8x^{3}+\cdots = 3
\]
\(x = \frac{1}
{3}\)
\(x = \frac{1}
{5}\)
\(x = \frac{1}
{2}\)
\(x = \frac{3}
{4}\)
9000062908
Level:
B
A quarter circle is an arc formed by one quarter of the full circle. An infinite spiral is
built from quarter circles with an increasing radius. The radius of the first quarter circle
is \(4\, \mathrm{cm}\).
The radius of each quarter circle in the spiral is one half of the radius of the previous
quarter circle. Find the total length of the spiral.
\(4\pi \)
\(8\)
\(\frac{8}
{3}\)
\(\infty \)
9000062909
Level:
B
Consider the square of the side \(4\, \mathrm{cm}\).
The second square is inscribed into this first square by joining the centers of all sides.
In a similar way, the third square is inscribed into the second square by joining the
centers of the sides of the second square and this process continues up to infinity.
Find the sum of the perimeters of all squares.
\(32 + 16\sqrt{2}\)
\(32 - 16\sqrt{2}\)
\(32\)
\(\infty \)
9000062910
Level:
B
Consider the square of the side \(4\, \mathrm{cm}\).
The second square is inscribed into this first square by joining the centers of all sides.
In a similar way, the third square is inscribed into the second square by joining the
centers of the sides of the second square and this process continues up to infinity.
Find the sum of the squares of all squares.
\(32\)
\(40\)
\(\frac{32}
{3} \)
\(\infty \)
9000062907
Level:
B
A quarter circle is an arc formed by one quarter of the full circle. An infinite spiral is
built from quarter circles with an increasing radius. The radius of the first quarter circle
is \(1\, \mathrm{cm}\).
The radius of each quarter circle in the spiral is bigger by one half than the radius of
the previous quarter circle. Find the total length of the spiral.
\(\infty \)
\(4\pi \)
\(\frac{2}
{5}\pi \)
\(\frac{1}
{3}\pi \)
9000062901
Level:
A
Find the sum of the following geometric series.
\[
-\frac{1}
{3} + \frac{1}
{6} - \frac{1}
{12} + \frac{1}
{24}-\cdots
\]
\(-\frac{2}
{9}\)
\(-\frac{2}
{3}\)
\(\frac{2}
{9}\)
\(\infty \)
9000062903
Level:
B
A semicircle is a half of the full circle. An infinite spiral is built from
semicircles with an increasing radius. The radius of the first semicircle is
\(3\, \mathrm{cm}\). The
radius of each semicircle in the spiral is bigger than the radius of the previous
semicircle by one third. Find the total length of the spiral.
\(\infty \)
\(9\pi \)
\(9\)
\(3\pi \)
9000062904
Level:
B
A semicircle is a half of the full circle. An infinite spiral is built from
semicircles with a decreasing radius. The radius of the first semicircle is
\(3\, \mathrm{cm}\). The
radius of each semicircle in the spiral is smaller by one third of the radius of the
previous semicircle. Find the total length of the spiral.
\(9\pi \)
\(9\)
\(\frac{9}
{5}\pi \)
\(\infty \)
9000062902
Level:
A
Find the sum of the following geometric series.
\[
1 + \frac{3}
{2} + \frac{9}
{4} + \frac{27}
{8} + \frac{81}
{16}+\cdots
\]
\(\infty \)
\(- 2\)
\(2\)
\(\frac{2}
{5}\)