U-0-1-U-1-2-U-n-2-3-2-U-n-1-1-2-U-n-Determinate-the-smallest-integer-n-0-such-that-n-n-0-we-have-U-n-3-10-4-

Question Number 133856 by mathocean1 last updated on 24/Feb/21

$$$$$$\{\begin{array}{l}{U}_0=1\\ U_1=2\\ U_{n+2}=\frac{3}{2}{U}_{n+1}-\frac{1}{2}{U}_n\end{array}$$$$Determinatethesmallestinteger$$$$n_{0}suchthat\mathrm{\forall }n⩾n_{0}wehave\mid U_n-3\mid ⩽{10}^{-4}$$

Answered by mr W last updated on 25/Feb/21

$$x^2-\frac{3}{2}x+\frac{1}{2}=0$$$$(x-1)(x-\frac{1}{2})=0$$$$x=1,\frac{1}{2}$$$$\Rightarrow U_n=A+\frac{B}{2^n}$$$$U_0=A+B=1$$$$U_1=A+\frac{B}{2}=2$$$$\Rightarrow B=-2,A=3$$$$\Rightarrow U_n=3-\frac{1}{2^{n-1}}$$$$\underset{n\to \mathrm{\infty }}{\lim }{U}_n=3$$$$\mid U_n-3\mid =\frac{1}{2^{n-1}}⩽{10}^{-4}$$$$2^{n-1}⩾{10}^4$$$$n⩾1+{\log }_2{10}^4=1+\frac{4}{\log 2}\approx 14.29$$$$\Rightarrow n_0=15$$

Commented by mathocean1 last updated on 27/Feb/21

$$wow{it}^{\prime }sgreatsir\dots$$$$Canyouexplainmehowyoufindthe$$$$firstandfourthlinesplease\dots$$

Commented by mr W last updated on 27/Feb/21

$$wecanuse“characteristicroot{method}^{″}$$$$tosolvethiskindofproblemswith$$$$“recurrence{relation}^{″}.youmayget$$$$someexplanationhere:$$$$(justreadthetextafterexample2.4.5.$$

Commented by mr W last updated on 27/Feb/21

Commented by mr W last updated on 27/Feb/21

https://math.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame_IN/SMC%3A_MATH_339_-_Discrete_Mathematics_(Rohatgi)/Text/2%3A_Sequences/2.4%3A_Solving_Recurrence_Relations

Commented by mathocean1 last updated on 27/Feb/21

$$Super!$$$$$$$$Thankyouverymuchsir\dots$$

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