Question Number 133856 by mathocean1 last updated on 24/Feb/21
$$ \\ $$$$\begin{cases}{{U}_{\mathrm{0}} =\mathrm{1}}\\{{U}_{\mathrm{1}} =\mathrm{2}}\\{\:{U}_{{n}+\mathrm{2}} =\frac{\mathrm{3}}{\mathrm{2}}{U}_{{n}+\mathrm{1}} −\frac{\mathrm{1}}{\mathrm{2}}{U}_{{n}} }\end{cases} \\ $$$${Determinate}\:{the}\:{smallest}\:{integer} \\ $$$${n}_{\mathrm{0}} \:{such}\:{that}\:\forall\:{n}\geqslant{n}_{\mathrm{0}} \:{we}\:{have}\:\mid{U}_{{n}} −\mathrm{3}\mid\leqslant\mathrm{10}^{−\mathrm{4}} \\ $$
Answered by mr W last updated on 25/Feb/21
$${x}^{\mathrm{2}} −\frac{\mathrm{3}}{\mathrm{2}}{x}+\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0} \\ $$$$\left({x}−\mathrm{1}\right)\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)=\mathrm{0} \\ $$$${x}=\mathrm{1},\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow{U}_{{n}} ={A}+\frac{{B}}{\mathrm{2}^{{n}} } \\ $$$${U}_{\mathrm{0}} ={A}+{B}=\mathrm{1} \\ $$$${U}_{\mathrm{1}} ={A}+\frac{{B}}{\mathrm{2}}=\mathrm{2} \\ $$$$\Rightarrow{B}=−\mathrm{2},\:{A}=\mathrm{3} \\ $$$$\Rightarrow{U}_{{n}} =\mathrm{3}−\frac{\mathrm{1}}{\mathrm{2}^{{n}−\mathrm{1}} } \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{U}_{{n}} =\mathrm{3} \\ $$$$\mid{U}_{{n}} −\mathrm{3}\mid=\frac{\mathrm{1}}{\mathrm{2}^{{n}−\mathrm{1}} }\leqslant\mathrm{10}^{−\mathrm{4}} \\ $$$$\mathrm{2}^{{n}−\mathrm{1}} \geqslant\mathrm{10}^{\mathrm{4}} \\ $$$${n}\geqslant\mathrm{1}+\mathrm{log}_{\mathrm{2}} \:\mathrm{10}^{\mathrm{4}} =\mathrm{1}+\frac{\mathrm{4}}{\mathrm{log}\:\mathrm{2}}\approx\mathrm{14}.\mathrm{29} \\ $$$$\Rightarrow{n}_{\mathrm{0}} =\mathrm{15} \\ $$
Commented by mathocean1 last updated on 27/Feb/21
$${wow}\:{it}'{s}\:{great}\:{sir}… \\ $$$${Can}\:{you}\:{explain}\:{me}\:{how}\:{you}\:{find}\:{the} \\ $$$${first}\:{and}\:{fourth}\:{lines}\:{please}… \\ $$
Commented by mr W last updated on 27/Feb/21
$${we}\:{can}\:{use}\:“{characteristic}\:{root}\:{method}'' \\ $$$${to}\:{solve}\:{this}\:{kind}\:{of}\:{problems}\:{with} \\ $$$$“{recurrence}\:{relation}''.\:{you}\:{may}\:{get} \\ $$$${some}\:{explanation}\:{here}: \\ $$$$\left({just}\:{read}\:{the}\:{text}\:{after}\:{example}\:\mathrm{2}.\mathrm{4}.\mathrm{5}.\right. \\ $$
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
$${Super}! \\ $$$$ \\ $$$${Thank}\:{you}\:\:{very}\:\:{much}\:{sir}\:… \\ $$