2010006805
Level:
A
The picture shows first three patterns formed by dots. The numbers of dots in patterns correspond to terms of an arithmetic sequence. To which term corresponds a pattern that will use \(28\) dots?
Nineth
Eighth
Seventh
Tenth
Arithmetic sequences
2010006804
Level:
A
In the picture there are the first four terms of the sequence of patterns that consist of black and white squares. Identify the true statement about the number of squares in these patterns, if you know that exactly one of the given statements is true.
The numbers of white squares in patterns determine an arithmetic sequence with the common difference \(4\).
The numbers of black squares in patterns determine an arithmetic sequence with the common difference \(4\).
The numbers of white squares in patterns determine an arithmetic sequence with the third term \(16\).
The numbers of black squares in patterns determine an arithmetic sequence with the third term \(16\).
2010006803
Level:
A
In the picture there are the first four terms of the sequence of patterns that consist of black and white squares. Identify the true statement about the number of squares in these patterns, if you know that exactly one of the given statements is true.
The numbers of black squares in patterns determine an arithmetic sequence with the common difference \(6\).
The numbers of white squares in patterns determine an arithmetic sequence with the common difference \(6\).
The numbers of white squares in patterns determine an arithmetic sequence with the first term \(2\).
The numbers of black squares in patterns determine an arithmetic sequence with the first term \(2\).
2010006802
Level:
B
We are given an arithmetic sequence \( (a_n)^{\infty}_{n=1}\), where \(a_3=328\), \(a_5=320\). Find \( n\), so that \(a_n=0\).
\(85\)
\(79\)
\(83\)
\(81\)
2010006801
Level:
B
We are given an arithmetic sequence \( (a_n)^{\infty}_{n=1}\), where \(a_2=429\), \(a_5=420\). Find \(n\), so that \(a_n=0\).
\(145\)
\(141\)
\(144\)
\(142\)
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 $\mathbb{N}$
2010000503
Level:
C
Find the sum of the integers which satisfy the following inequality.
\[
x^{2} + 8x - 153\leq 0
\]
\( -108\)
\( 108\)
\( 104\)
\( -104\)
\( -100\)
2010000502
Level:
B
Suppose three numbers form three consecutive terms of an arithmetic sequence. The sum of these numbers is \(36\) and the product is \(-972\). Find the biggest of these three numbers.
\( 27\)
\( 12\)
\( 17\)
\( -3\)
\( 15\)
2010000501
Level:
B
Find the sum of the first $15$ negative terms of the arithmetic sequence: $\left(7-2n\right)_{n=1}^{\infty}$.
\( -225\)
\( -135\)
\( -240\)
\( -180\)
\( -450\)
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