Introduction to sequences

2010000404

Level: 
A
Which sequence is defined by the given graph?
\( \left( a_n \right)^{5}_{n=1} = 3,\ \ 2,\ \ 1,\ \ 2,\ \ 3 \)
\( \left( a_n \right)^{10}_{n=1} = 1,\ \ 3,\ \ 2,\ \ 2,\ \ 3,\ \ 1,\ \ 4,\ \ 2,\ \ 5,\ \ 3 \)
\( \left( a_n \right)^{5}_{n=1} = 1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \)
\( \left( a_n \right)^{5}_{n=1} = 1,\ \ 2,\ \ 2,\ \ 3,\ \ 3 \)

2010000403

Level: 
A
We are given a sequence \( \left( 5n-3\right)^{\infty}_{n=1} \). What does this formula express?
a sequence of all natural numbers which after dividing by \(5\) give the remainder \(2\)
a sequence of all natural numbers which are divisible by \(3\)
a sequence of all natural numbers which are divisible by \(5\)
a sequence of all natural numbers which after dividing by \(5\) give the remainder \(3\)

2010000402

Level: 
B
We are given the sequence \( \left( \frac{n}{n+1} \right)^{\infty}_{n=1} \). Find the recursive formula of such sequence.
\( a_1=\frac{1}{2}\,;\ a_{n+1}=a_n\frac{(n+1)^2}{n(n+2)},\ n\in\mathbb{N} \)
\( a_1={2}\,;\ a_{n+1}=a_n\frac{(n+1)^2}{n(n+2)},\ n\in\mathbb{N} \)
\( a_1=\frac{1}{2}\,;\ a_{n+1}=a_n\frac{n(n+1)}{(n+1)(n+2)},\ n\in\mathbb{N} \)
\( a_1={2}\,;\ a_{n+1}=a_n\frac{n(n+1)}{(n+1)(n+2)},\ n\in\mathbb{N} \)

2010000401

Level: 
A
We are given a sequence \( \left( \frac{n}{n+1} \right)_{n=1}^{\infty} \). Which of the following formulations describes how is the given sequence defined?
defined by a formula for the \(n\)th term
defined by a list of the sequence elements
defined by a graph of the sequence
defined by a recursive formula for the sequence

1003084909

Level: 
B
We were given an oscillating sequence \( 3\text{, }-3\text{, }\ 3\text{, }-3\text{, }\ 3\dots \) (numbers \( 3 \) and \( -3 \) alternate regularly). What is the formula for the $n$th element of the sequence?
\( a_n=(-1)^{n+1}\cdot3\text{, }\ n\in\mathbb{N} \)
\( a_n=(-1)^{n}\cdot3\text{, }\ n\in\mathbb{N} \)
\( a_n=3^n\text{, }\ n\in\mathbb{N} \)
\( a_n=-3^n\text{, }\ n\in\mathbb{N} \)

1003084907

Level: 
A
A sequence \( \left( a_n \right)^{\infty}_{n=1} \) is defined by the relations: \( a_1=3;\ a_{n+1}=\frac{a_n}{n+2}\text{, }n\in\mathbb{N} \). Which of the following descriptions is used to define the given sequence?
a recursion formula of a sequence
a formula of the \(n\)th term
a list of sequence elements
a graph of a sequence

1003084906

Level: 
A
We are given a sequence \( \left( \frac{n+1}n \right)_{n=1}^{\infty} \). Which of the following formulations describes how is the given sequence defined?
defined by a formula for the \( n \)th term
defined by a list of the sequence elements
defined by a graph of the sequence
defined by a recursive formula for the sequence

1103084905

Level: 
A
Which sequence is defined by the given graph?
\( \left( a_n \right)^{5}_{n=1} = 2\text{, }\ 1\text{, }\ 3\text{, }\ 1\text{, }\ 2 \)
\( \left( a_n \right)^{10}_{n=1} = 1\text{, }\ 2\text{, }\ 2\text{, }\ 1\text{, }\ 3\text{, }\ 3\text{, }\ 4\text{, }\ 1\text{, }\ 5\text{, }\ 2 \)
\( \left( a_n \right)^{5}_{n=1} = 1\text{, }\ 2\text{, }\ 3\text{, }\ 4\text{, }\ 5 \)
\( \left( a_n \right)^{5}_{n=1} = 1\text{, }\ 1\text{, }\ 2\text{, }\ 2\text{, }\ 3 \)

1003084904

Level: 
A
We were given a sequence \( \left( \frac1n \right)^{\infty}_{n=1} \). Which expression describes the term \( a_{n-1}\ (n\in\mathbb{N}; n>1 ) \) of the given sequence?
\( a_{n-1} = \frac1{n-1} \)
\( a_{n-1} = n-1 \)
\( a_{n-1} = \frac1n \)
\( a_{n-1} = \frac1{2n-1} \)