2010000704
Level:
B
The \(n\)th term of a sequence is given by \(a_n = \frac{n^2-1}{n+5}\). Which term of this sequence is equal to \(4\)?
the seventh term
the third term
the twenty-first term
the fourth term
Introduction to sequences
2010000703
Level:
B
Consider the sequence \(2a_{n} = a_{n+1} - a_{n-1}\)
with \(a_{3} = 2\)
and \(a_{4} = 5\).
Find \(a_{2} - a_{1}\).
\( 1\)
\(4\)
\(-20\)
\(-25\)
2010000702
Level:
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \( a_1=-1,\ a_2=0\) and \(\ a_{n+2}=a_{n}-a_{n+1}-d\), where \(\ n\in\mathbb{N} \).
Find the value of an unknown constant \( d\in\mathbb{R} \) and of the term \( a_5 \) if \( a_3 = -4 \).
\( d=3,\ a_5=-8 \)
\( d=5,\ a_5=-10 \)
\( d=3,\ a_5=1\)
\( d=5,\ a_5=-9 \)
2010000701
Level:
B
We are given the sequence \(\left (an + b\right )_{n=1}^{\infty }\).
This sequence satisfies \(a_{7} - a_{2} = -10\). Use this information to find \(a\).
\(a = -2\)
\(a = 2\)
\(a = -1\)
\(a = 1\)
2010000406
Level:
A
We are given a sequence \( \left( a_n \right)^{5}_{n=1}\) defined by the following graph. Find the formula of the \(n\)th term of this sequence.
\( a_n = 2n-3,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 2n,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 3-2n,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 2n-1,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
2010000405
Level:
A
We are given a sequence \( \left( a_n \right)^{5}_{n=1}\) defined by the following graph. Find the formula of the \(n\)th term of this sequence.
\( a_n = 3-2n,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 2n,\ n\in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 1-2n,\ n\in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 2n-3,\ n\in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
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