1003107209
Level:
C
Solve the equation with the real variable $x$:
$$ \sum\limits_{n=1}^{23} xn=92 $$
$\frac13$
$4$
$\frac14$
$1$
$\frac12$
Arithmetic sequences
1003107210
Level:
C
Solve the inequality with the variable $x$, where $x$ is a natural number:
$$\sum\limits_{n=1}^x \left(\frac25n-1\right) \leq x $$
$ x\leq9;\ x\in\mathbb{N}$
$x\leq10;\ x\in\mathbb{N}$
$x\in(-\infty;8]$
$x\geq8;\ x\in\mathbb{N}$
there is no solution in $\mathbb{N}$
2010000209
Level:
C
Insert \(3\) numbers between the roots of the equation \( 4x^2 -17x+4=0\) such that the resulting sequence is a part (\(5\) terms) of an increasing arithmetic sequence with the common difference \(d\). Choose the incorrect statement about the common difference \(d\).
\(d\) is a fraction greater than \(1\)
\( d>0\)
\(d\) is a fraction smaller than \(1\)
\(d\) is a rational number
2010000210
Level:
C
Insert \(3\) numbers between the roots of the equation \( 5x^2 -26x+5=0\) such that the resulting sequence is a part (\(5\) terms) of an increasing arithmetic sequence with the common difference \(d\). Choose the incorrect statement about the common difference \(d\).
\(d\) is a fraction smaller than \(1\)
\( d>0\)
\(d\) is a fraction greater than \(1\)
\(d\) is a rational number
2010000503
Level:
C
Find the sum of the integers which satisfy the following inequality.
\[
x^{2} + 8x - 153\leq 0
\]
\( -108\)
\( 108\)
\( 104\)
\( -104\)
\( -100\)
2010000504
Level:
C
Solve the inequality with the natural variable $x$:
$$\sum\limits_{n=1}^x \left(5-\frac12n\right) \geq x $$
$ x\leq 15;\ x\in\mathbb{N}$
$ x\geq 15;\ x\in\mathbb{N}$
$ x\leq 17;\ x\in\mathbb{N}$
$x\in(-\infty;15]$
there is no solution in $\mathbb{N}$
9000064801
Level:
C
The sides of a right triangle form three consecutive terms of an arithmetic sequence. The perimeter
of the triangle is \(60\, \mathrm{cm}\).
Find the hypotenuse of the triangle (the longest side).
\(25\, \mathrm{cm}\)
\(12\, \mathrm{cm}\)
\(15\, \mathrm{cm}\)
\(20\, \mathrm{cm}\)
\(30\, \mathrm{cm}\)
9000064802
Level:
C
The arithmetic sequence is given by the third term
\(a_{3} = 5\) and the common
difference \(d = 2\).
How many terms of the sequence has to be summed up to ensure that the sum is bigger
than \(300\)?
\(18\)
\(10\)
\(12\)
\(14\)
\(16\)
9000064803
Level:
C
Suppose three numbers form three consecutive terms of an arithmetic sequence. The sum of these
numbers is \(33\) and
the product is \(1\: 155\).
Find the smallest of these three numbers.
\(7\)
\(9\)
\(11\)
\(13\)
\(15\)
9000064804
Level:
C
An arithmetic sequence is formed by consecutive odd integers. The twelfth term is
\( 53\). Find
the sum of the first five terms.
\(175\)
\(151\)
\(163\)
\(187\)
\(199\)