Kenneth Rosen Edition 7 Exercise 2.4 Question 3 (Page No. 167)

Question

What are the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ of the sequence $\{a_{n}\},$ where $a_{n}$ equals

• A. $2^{n} + 1$
• B. $(n + 1)^{n+1}$
• C. $\left \lfloor n/2\right \rfloor$
• D. $\left \lfloor n/2\right \rfloor + \left \lceil n/2\right \rceil$

Answer 1

A).$2^{n}+1$

$n=0,2^{0}+1=1+1=2$

$n=1,2^{1}+1=2+1=3$

$n=2,2^{2}+1=4+1=5$

$n=3,2^{3}+1=8+1=9$

B).$(n+1)^{n+1}$

$n=0,(0+1)^{0+1}=1^{1}=1$

$n=1,(1+1)^{1+1}=2^{2}=4$

$n=2,(2+1)^{2+1}=3^{3}=27$

$n=3,(3+1)^{3+1}=4^{4}=64$

C).$⌊n/2⌋$

$n=0,⌊0/2⌋=0$

$n=1,⌊1/2⌋=0$

$n=2,⌊2/2⌋=1$

$n=3,⌊3/2⌋=1$

D).$⌊n/2⌋+⌈n/2⌉$

$n=0,⌊0/2⌋+⌈0/2⌉=0+0=0$

$n=1,⌊1/2⌋+⌈1/2⌉=0+1=1$

$n=2,⌊2/2⌋+⌈2/2⌉=1+1=2$

$n=3,⌊3/2⌋+⌈3/2⌉=1+2=3$