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Question-88471




Question Number 88471 by M±th+et£s last updated on 10/Apr/20
Answered by mr W last updated on 10/Apr/20
a_(n+1) =2+(5/a_n )  lim_(n→∞) a_(n+1) =lim_(n→∞) (2+(5/a_n ))  l=2+(5/l)  l^2 −2l−5=0  ⇒l=1+(√6)
$${a}_{{n}+\mathrm{1}} =\mathrm{2}+\frac{\mathrm{5}}{{a}_{{n}} } \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{a}_{{n}+\mathrm{1}} =\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{2}+\frac{\mathrm{5}}{{a}_{{n}} }\right) \\ $$$${l}=\mathrm{2}+\frac{\mathrm{5}}{{l}} \\ $$$${l}^{\mathrm{2}} −\mathrm{2}{l}−\mathrm{5}=\mathrm{0} \\ $$$$\Rightarrow{l}=\mathrm{1}+\sqrt{\mathrm{6}} \\ $$
Commented by M±th+et£s last updated on 10/Apr/20
thanx sir but why lim_(n→∞) a_(n+1) =a_n
$${thanx}\:{sir}\:{but}\:{why}\:\underset{{n}\rightarrow\infty} {{lim}a}_{{n}+\mathrm{1}} ={a}_{{n}} \\ $$
Commented by M±th+et£s last updated on 10/Apr/20
sorry i mean lim_(n→∞)  a_n
$${sorry}\:{i}\:{mean}\:\underset{{n}\rightarrow\infty} {{lim}}\:{a}_{{n}} \\ $$
Commented by M±th+et£s last updated on 11/Apr/20
and can we find a_n
$${and}\:{can}\:{we}\:{find}\:{a}_{{n}} \\ $$
Commented by mr W last updated on 11/Apr/20
if convergence exists, i.e.  lim_(n→∞) a_n =l, that means when n→∞,  a_n →l  then it′s clear, when n→∞, ⇒n+2→∞  ⇒a_(n+2) →l, therefore:  lim_(n→∞) a_(n+2) =l  lim_(n→∞) a_(n+2000000) =l  lim_(n→∞) a_(n−100) =l
$${if}\:{convergence}\:{exists},\:{i}.{e}. \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{a}_{{n}} ={l},\:{that}\:{means}\:{when}\:{n}\rightarrow\infty, \\ $$$${a}_{{n}} \rightarrow{l} \\ $$$${then}\:{it}'{s}\:{clear},\:{when}\:{n}\rightarrow\infty,\:\Rightarrow{n}+\mathrm{2}\rightarrow\infty \\ $$$$\Rightarrow{a}_{{n}+\mathrm{2}} \rightarrow{l},\:{therefore}: \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{a}_{{n}+\mathrm{2}} ={l} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{a}_{{n}+\mathrm{2000000}} ={l} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{a}_{{n}−\mathrm{100}} ={l} \\ $$
Commented by mr W last updated on 11/Apr/20
i′m not sure that we can get the explicit  form for a_n .
$${i}'{m}\:{not}\:{sure}\:{that}\:{we}\:{can}\:{get}\:{the}\:{explicit} \\ $$$${form}\:{for}\:{a}_{{n}} . \\ $$
Commented by M±th+et£s last updated on 11/Apr/20
thank you sir
$${thank}\:{you}\:{sir} \\ $$

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