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Question Number 206351 by MetaLahor1999 last updated on 12/Apr/24
expression of the sequence (a_n ) defined by { ((a_0 >0 , a_1 >0)),((a_(n+2) =((2(−1)^n )/(n+2))−((2(−1)^n (2n+3))/(n+2))a_(n+1) +((n+1)/(n+2))a_n )) :}
$${expression}\:{of}\:{the}\:{sequence}\:\left({a}_{{n}} \right)\:{defined} \\ $$$${by}\: \\ $$$$\begin{cases}{{a}_{\mathrm{0}} >\mathrm{0}\:,\:{a}_{\mathrm{1}} >\mathrm{0}}\\{{a}_{{n}+\mathrm{2}} =\frac{\mathrm{2}\left(−\mathrm{1}\right)^{{n}} }{{n}+\mathrm{2}}−\frac{\mathrm{2}\left(−\mathrm{1}\right)^{{n}} \left(\mathrm{2}{n}+\mathrm{3}\right)}{{n}+\mathrm{2}}{a}_{{n}+\mathrm{1}} +\frac{{n}+\mathrm{1}}{{n}+\mathrm{2}}{a}_{{n}} }\end{cases} \\ $$
Commented by TheHoneyCat last updated on 12/Apr/24
⇔ { ((a_0 >0 , a_1 >0)),((a_(n+2) =((2(−1)^n (1−2n−3))/(n+2))a_(n+1) +((n+1)/(n+2))a_n )) :} ⇔ { ((a_0 >0 , a_1 >0)),((a_(n+2) =((4(−1)^(n+1) (n+1))/(n+2))a_(n+1) +((n+1)/(n+2))a_n )) :} ⇔ { ((a_0 >0 , a_1 >0)),((a_(n+2) =((n+1)/(n+2))(4(−1)^(n+1) a_(n+1) +a_n ))) :} It is not a solution in any way... it′s just that I think this reformulation might make it easier... I might continue to work on it tomorow... good luck to anyone attempting.
$$\Leftrightarrow\begin{cases}{{a}_{\mathrm{0}} >\mathrm{0}\:,\:{a}_{\mathrm{1}} >\mathrm{0}}\\{{a}_{{n}+\mathrm{2}} =\frac{\mathrm{2}\left(−\mathrm{1}\right)^{{n}} \left(\mathrm{1}−\mathrm{2}{n}−\mathrm{3}\right)}{{n}+\mathrm{2}}{a}_{{n}+\mathrm{1}} +\frac{{n}+\mathrm{1}}{{n}+\mathrm{2}}{a}_{{n}} }\end{cases} \\ $$$$\Leftrightarrow\begin{cases}{{a}_{\mathrm{0}} >\mathrm{0}\:,\:{a}_{\mathrm{1}} >\mathrm{0}}\\{{a}_{{n}+\mathrm{2}} =\frac{\mathrm{4}\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \left({n}+\mathrm{1}\right)}{{n}+\mathrm{2}}{a}_{{n}+\mathrm{1}} +\frac{{n}+\mathrm{1}}{{n}+\mathrm{2}}{a}_{{n}} }\end{cases} \\ $$$$\Leftrightarrow\begin{cases}{{a}_{\mathrm{0}} >\mathrm{0}\:,\:{a}_{\mathrm{1}} >\mathrm{0}}\\{{a}_{{n}+\mathrm{2}} =\frac{{n}+\mathrm{1}}{{n}+\mathrm{2}}\left(\mathrm{4}\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} {a}_{{n}+\mathrm{1}} +{a}_{{n}} \right)}\end{cases} \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{solution}\:\mathrm{in}\:\mathrm{any}\:\mathrm{way}… \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{just}\:\mathrm{that}\:\mathrm{I}\:\mathrm{think}\:\mathrm{this}\:\mathrm{reformulation}\:\mathrm{might} \\ $$$$\mathrm{make}\:\mathrm{it}\:\mathrm{easier}… \\ $$$$\mathrm{I}\:\mathrm{might}\:\mathrm{continue}\:\mathrm{to}\:\mathrm{work}\:\mathrm{on}\:\mathrm{it}\:\mathrm{tomorow}… \\ $$$$\mathrm{good}\:\mathrm{luck}\:\mathrm{to}\:\mathrm{anyone}\:\mathrm{attempting}. \\ $$

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