Introduction to sequences

1003085002

Level: 
A
We were given the sequence \( \left(\frac{n+3}{2n}\right)_{n=1}^{\infty} \). What are its first five terms?
\( 2 \), \( \frac54 \), \( 1 \), \( \frac78 \), \( \frac45 \)
\( \frac45 \), \( \frac78 \), \( 1 \), \( \frac54 \), \( 2 \)
\( 2 \), \( \frac45 \), \( 1 \), \( \frac87 \), \( \frac54 \)
\( \frac12 \), \( \frac23 \), \( \frac34 \), \( \frac45 \), \( \frac56 \)

1003085001

Level: 
A
We were given the sequence \( \left(\frac1{3^n}\right)_{n=1}^{\infty} \). What are its first five terms?
\( \frac13 \), \( \frac19 \), \( \frac1{27} \), \( \frac1{81} \), \( \frac1{243} \)
\( 3 \), \( 9 \), \( 27 \), \( 81 \), \( 243 \)
\( 3 \), \( 6 \), \( 9 \), \( 12 \), \( 15 \)
\( \frac13 \), \( \frac16 \), \( \frac19 \), \( \frac1{12} \), \( \frac1{15} \)

1003107310

Level: 
A
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \( a_1=1,\ a_2=2\,;\ a_{n+2}=\frac12\left( a_{n+1}+a_n\right),\ n\in\mathbb{N} \). Find the sum of the first four terms of this sequence.
\( \frac{25}4 \)
\( \frac{63}8 \)
\( \frac{13}4 \)
\( \frac4{25} \)

1003107309

Level: 
B
We are given the sequence \( \left(\log2^n \right)^{\infty}_{n=1} \). Find the recursive formula of such sequence.
\( a_1=\log 2\,;\ a_{n+1}=a_n+\log 2,\ n\in\mathbb{N} \)
\( a_1=\log 2\,;\ a_{n+1}=a_n\cdot\log 2,\ n\in\mathbb{N} \)
\( a_1=\log 2\,;\ a_{n+1}=a_n-\log 2,\ n\in\mathbb{N} \)
\( a_1=\log 2\,;\ a_{n+1}=a_n+\log 2^n,\ n\in\mathbb{N} \)

1003107307

Level: 
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \( a_1=1,\ a_2=5\) and \(\ a_{n+2}=a_{n+1}-a_n+d\), where \(\ n\in\mathbb{N} \). Find the value of an unknown constant \( d\in\mathbb{R} \) and of the term \( a_5 \) if \( a_3 = 10 \).
\( d=6,\ a_5=7 \)
\( d=6,\ a_5=6 \)
\( d=7,\ a_5=6 \)
\( d=7,\ a_5=7 \)

1003107306

Level: 
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \( a_1=1\,;\ a_{n+1}=2a_n,\ n\in\mathbb{N} \). Find the \( n \)th term of this sequence.
\( a_n=2^{n-1},\ n\in\mathbb{N} \)
\( a_n=2^n,\ n\in\mathbb{N} \)
\( a_n=2^{n+1},\ n\in\mathbb{N} \)
\( a_n=2^n-1,\ n\in\mathbb{N} \)

1003107305

Level: 
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \( a_1=5\,;\ a_{n+1}=a_n+4,\ n\in\mathbb{N} \). Find the \( n \)th term of this sequence.
\( a_n=4n+1,\ n\in\mathbb{N} \)
\( a_n=4n-1,\ n\in\mathbb{N} \)
\( a_n=4n,\ n\in\mathbb{N} \)
\( a_n=5n,\ n\in\mathbb{N} \)

1003107304

Level: 
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \(a_1=0\,;\ a_{n+1}=2-a_n,\ n\in\mathbb{N} \). Find the \( n \)th term of this sequence.
\( a_n=1+(-1)^n,\ n\in\mathbb{N} \)
\( a_n=1+(-1)^{n+1},\ n\in\mathbb{N} \)
\( a_n=1+(-1)^{n-1},\ n\in\mathbb{N} \)
\( a_n=1-1^n,\ n\in\mathbb{N} \)

1003107303

Level: 
B
We are given the sequence \( \left( 2^{2-n} \right)^{\infty}_{n=1} \). Find the recursive formula of such sequence.
\( a_1=2\,;\ a_{n+1}=a_n\cdot\frac12,\ n\in\mathbb{N} \)
\( a_1=2\,;\ a_{n+1}=a_n\cdot2,\ n\in\mathbb{N} \)
\( a_1=2\,;\ a_{n+1}=a_n\cdot2^n,\ n\in\mathbb{N} \)
\( a_1=2\,;\ a_{n+1}=a_n\cdot\frac1{2^n},\ n\in\mathbb{N} \)

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.
\( a_1=\frac{\sqrt2}4\left(\sqrt2-1\right);\ a_{n+1}=a_n\left(\sqrt2-1\right),\ n\in\mathbb{N} \)
\( a_1=\frac{\sqrt2}4;\ a_{n+1}=a_n\left(\sqrt2-1\right),\ n\in\mathbb{N} \)
\( a_1=\sqrt2-1\,;\ a_{n+1}=a_n\left(\sqrt2-1\right),\ n\in\mathbb{N} \)
\( a_1=\frac{\sqrt2}4\left(\sqrt2-1\right);\ a_{n+1}=a_n\left(\sqrt2-1\right)^2,\ n\in\mathbb{N} \)