Introduction to sequences

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.
\( a_n = 3-2n,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 2n,\ n\in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 1-2n,\ n\in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 2n-3,\ n\in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)

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.
\( a_n = 2n-3,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 2n,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 3-2n,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)
\( a_n = 2n-1,\ n \in\{1,\ \ 2,\ \ 3,\ \ 4,\ \ 5 \}\)

9000063810

Level: 
A
Consider the sequences \(\left (a_{n}\right )_{n=1}^{\infty }\) and \(\left (b_{n}\right )_{n=1}^{\infty }\) where \(a_{n} = 2^{n}\) and \(b_{n} = n^{2} - 1\), respectively. Identify a true statement in the terms of these sequences.
\(a_{3} = b_{3}\)
\(a_{2} = b_{2} + 2\)
\(a_{4} = b_{4} - 2\)
\(a_{5} = b_{5} - 8\)

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?
\( a_n=(-1)^{n+1}\cdot3\text{, }\ n\in\mathbb{N} \)
\( a_n=(-1)^{n}\cdot3\text{, }\ n\in\mathbb{N} \)
\( a_n=3^n\text{, }\ n\in\mathbb{N} \)
\( a_n=-3^n\text{, }\ n\in\mathbb{N} \)

1003107301

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