Answer to Question #336143 in Discrete Mathematics for habibi

Question

1.   The sequence {Un} is defined by the first term u1=-2 and the recurrent relation : Un+1=Un/4 +3

a.   Evaluate U2,U3,U4

b.    Consider  the sequence Vn such that Vn=Un+1-Un for n>=2

c.    Prove that {Vn} is a G.P 

d.   Express Vn  and Un  in terms of n.

e.     Evaluate the limits of  Un and Vn  

f.    Is Un an increasing  or decreasing sequence ? 

Answer 1

a.

$$u_1=-2,$$

$$u_2=-2/4+3=\dfrac{5}{2},$$

$$u_3=(\dfrac{5}{2})/4+3=\dfrac{29}{8},$$

$$u_4=(\dfrac{29}{8})/4+3=\dfrac{125}{32}$$

b.

$$v_n=u_{n+1}-u_n, n\ge2$$

$$v_n=u_n/4+3-u_n, n\ge2$$

$$v_n=3-3u_n/4, n\ge2$$

c.

$$v_{n+1}=3-3u_{n+1}/4=3-3(u_n/4 +3)/4=$$

$$=(3-3u_n/4)/4=v_n/4, n\ge2$$

$\{v_n\}$ is a G.P 

$v_2=\dfrac{9}{8}, q=\dfrac{1}{4}$

d.

$$v_n=\dfrac{9}{8}(\dfrac{1}{4})^{n-2}, n\ge2$$

$$u_n=4-\dfrac{4v_n}{3}, n\ge2$$

$$u_n=4-\dfrac{3}{2}(\dfrac{1}{4})^{n-2}, n\ge2$$

$$u_1=-2$$

e.

$$\lim\limits_{n\to\infin}u_n=\lim\limits_{n\to\infin}(4-\dfrac{3}{2}(\dfrac{1}{4})^{n-2})=4$$

$$\lim\limits_{n\to\infin}v_n=\lim\limits_{n\to\infin}(\dfrac{9}{8}(\dfrac{1}{4})^{n-2})=0$$

f.

$\{u_n\}$ is increasing sequence.