Number Sequences: Problems with Solutions

Solution:
[tex]a_{n+1}-a_n=0[/tex] means that [tex]a_{n+1}=a_n[/tex], therefore [tex]137=a_1=a_2=a_3=...=a_{8999}[/tex]

Solution:
The recurrence relation is [tex]a_{n+2}-2a_{n+1}+a_n=0[/tex] and its characteristic equation is
[tex]r^2-2r+1=0[/tex] with a double root [tex]r_{1,2}=1[/tex]. It means that [tex]a_n=c_1.1^n+c_2.n.1^n=c_1+n.c_2[/tex]. But any sequence whose explicit member formula is a linear function is an arithmetic progression. Therefore [tex]a_n[/tex] is the arithmetic progression, defined by [tex]a_1=4[/tex] and [tex]d=a_2-a_1=6-4=2[/tex]. Then [tex]a_{15}=a_1+14.d=4+14.2=32[/tex]

Solution:
[tex]a_{19}=\frac{4.19-1}{19}=\frac{75}{19}[/tex]

Solution:
[tex]x_2=\frac{2.2+8}{4+3.2}=\frac{12}{10}=\frac{6}{5}[/tex]

Solution:
[tex]a_5==-3+2^5+5^{5-3}=-3+32+5^2=29+25=54[/tex]

Solution:
[tex]a_{n+1}-a_n=2^{n+1}+n+1-(2^n+n)=2.2^n+n+1-2^n-n=2^n+1>0[/tex], so the sequence is increasing.