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Question Number 2322 by prakash jain last updated on 15/Nov/15
a_0 =x  a_(n+1) =(1/(1+a_n ))  Find possible value for x such that  a_n =−1 for some n∈N.  For example:   x=−2, a_1 =−1  x=((−3)/2), a_2 =−1  x=? if a_n =−1
$${a}_{\mathrm{0}} ={x} \\ $$$${a}_{{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{1}+{a}_{{n}} } \\ $$$$\mathrm{Find}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{for}\:{x}\:\mathrm{such}\:\mathrm{that} \\ $$$${a}_{{n}} =−\mathrm{1}\:\mathrm{for}\:\mathrm{some}\:{n}\in\mathbb{N}. \\ $$$$\mathrm{For}\:\mathrm{example}:\: \\ $$$${x}=−\mathrm{2},\:{a}_{\mathrm{1}} =−\mathrm{1} \\ $$$${x}=\frac{−\mathrm{3}}{\mathrm{2}},\:{a}_{\mathrm{2}} =−\mathrm{1} \\ $$$${x}=?\:\mathrm{if}\:{a}_{{n}} =−\mathrm{1} \\ $$
Answered by 123456 last updated on 15/Nov/15
a_(n+1) =(1/(1+a_n ))  a_(n+1) +a_(n+1) a_n =1  a_n =((1−a_(n+1) )/a_(n+1) ),a_(n+1) ≠0  b_(n+1) =((1−b_n )/b_n ),b_1 =−1  b_2 =((1−−1)/(−1))=(2/(−1))=−2  b_3 =((1−−2)/(−2))=−(3/2)  b_4 =((1−−(3/2))/(−(3/2)))=−(5/3)  ∙∙∙  the sequence b_n  gives the x you want  b_n =^? −(F_(n+1) /F_n ) (check it)
$${a}_{{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{1}+{a}_{{n}} } \\ $$$${a}_{{n}+\mathrm{1}} +{a}_{{n}+\mathrm{1}} {a}_{{n}} =\mathrm{1} \\ $$$${a}_{{n}} =\frac{\mathrm{1}−{a}_{{n}+\mathrm{1}} }{{a}_{{n}+\mathrm{1}} },{a}_{{n}+\mathrm{1}} \neq\mathrm{0} \\ $$$${b}_{{n}+\mathrm{1}} =\frac{\mathrm{1}−{b}_{{n}} }{{b}_{{n}} },{b}_{\mathrm{1}} =−\mathrm{1} \\ $$$${b}_{\mathrm{2}} =\frac{\mathrm{1}−−\mathrm{1}}{−\mathrm{1}}=\frac{\mathrm{2}}{−\mathrm{1}}=−\mathrm{2} \\ $$$${b}_{\mathrm{3}} =\frac{\mathrm{1}−−\mathrm{2}}{−\mathrm{2}}=−\frac{\mathrm{3}}{\mathrm{2}} \\ $$$${b}_{\mathrm{4}} =\frac{\mathrm{1}−−\frac{\mathrm{3}}{\mathrm{2}}}{−\frac{\mathrm{3}}{\mathrm{2}}}=−\frac{\mathrm{5}}{\mathrm{3}} \\ $$$$\centerdot\centerdot\centerdot \\ $$$$\mathrm{the}\:\mathrm{sequence}\:{b}_{{n}} \:\mathrm{gives}\:\mathrm{the}\:{x}\:\mathrm{you}\:\mathrm{want} \\ $$$${b}_{{n}} \overset{?} {=}−\frac{\mathrm{F}_{{n}+\mathrm{1}} }{\mathrm{F}_{{n}} }\:\left(\mathrm{check}\:\mathrm{it}\right) \\ $$

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