Introduction to sequences

1003107302

Level: 
B
We are given the sequence \( \left( \frac{\sqrt2}4\left( \sqrt2 -1 \right)^n \right)^{\infty}_{n=1} \). Find the recursive formula of such sequence.
\( a_1=\frac{\sqrt2}4\left(\sqrt2-1\right);\ a_{n+1}=a_n\left(\sqrt2-1\right),\ n\in\mathbb{N} \)
\( a_1=\frac{\sqrt2}4;\ a_{n+1}=a_n\left(\sqrt2-1\right),\ n\in\mathbb{N} \)
\( a_1=\sqrt2-1\,;\ a_{n+1}=a_n\left(\sqrt2-1\right),\ n\in\mathbb{N} \)
\( a_1=\frac{\sqrt2}4\left(\sqrt2-1\right);\ a_{n+1}=a_n\left(\sqrt2-1\right)^2,\ n\in\mathbb{N} \)

1003107301

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

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.
\(a_{3} = b_{3}\)
\(a_{2} = b_{2} + 2\)
\(a_{4} = b_{4} - 2\)
\(a_{5} = b_{5} - 8\)