Find-x-n-n-Z-satisfying-x-0-0-x-1-1-and-x-n-1-x-n-x-n-1-2-1-x-n-1-x-n-2-1-for-n-1-

Question Number 7532 by Yozzia last updated on 02/Sep/16

F i n d x_{n} (n \in Z) s a t i s f y i n g x_{0} = 0, x_{1} = 1 a n d

x_{n + 1} = x_{n} \sqrt{x_{n - 1}^{2} + 1} + x_{n - 1} \sqrt{x_{n}^{2} + 1} f o r n ⩾ 1 .

Commented by sou1618 last updated on 03/Sep/16

w h e n n = 1

x_{2} = 1

w h e n n = 2

x_{3} = 2 \sqrt{2}

w h e n n = 3

x_{4} = 2 \sqrt{2} \sqrt{2} + \sqrt{8 + 1} = 7

\dots

w h e n n > 2 x_{n}, x_{n - 1} > 1

x_{n + 1} = x_{n} \sqrt{x_{n - 1}^{2} + 1} + x_{n - 1} \sqrt{x_{n}^{2} + 1} \cdot \cdot \cdot (*)

x_{n} = \frac{1}{2} (y_{n} - \frac{1}{y_{n}}) (y_{n} ⩾ 1)

\frac{1}{2} (y_{n + 1} - y_{n + 1}^{- 1}) = \frac{1}{2} (y_{n} - y_{n}^{- 1}) \frac{1}{2} (y_{n - 1} + y_{n - 1}^{- 1}) + \frac{1}{2} (y_{n - 1} - y_{n - 1}^{- 1}) \frac{1}{2} (y_{n} + y_{n}^{- 1})

2 (y_{n + 1} - y_{n + 1}^{- 1}) = (y_{n} - y_{n}^{- 1}) (y_{n - 1} + y_{n - 1}^{- 1}) + (y_{n - 1} - y_{n - 1}^{- 1}) (y_{n} + y_{n}^{- 1})

2 (y_{n + 1} - y_{n + 1}^{- 1}) = 2 y_{n} y_{n - 1} - 2 y_{n}^{- 1} y_{n}^{- 1}

y_{n + 1} - \frac{1}{y_{n + 1}} = y_{n} y_{n - 1} - \frac{1}{y_{n} y_{n - 1}}

y_{n + 1}^{2} - (y_{n} y_{n - 1} - \frac{1}{y_{n} y_{n - 1}}) y_{n + 1} - 1 = 0

(y_{n + 1} - y_{n} y_{n - 1}) (y_{n + 1} + \frac{1}{y_{n} y_{n - 1}}) = 0

y_{n + 1} = y_{n} y_{n - 1} (∵ y_{n} > 0)

x_{0} = 0, y_{0}^{2} - 0 y_{0} - 1 = 0

y_{0} = 1

x_{1} = 1, y_{1}^{2} - 2 y_{1} - 1 = 0

y_{1} = 1 + \sqrt{2}

y_{n} = y_{n - 1} y_{n - 2} = y_{n - 2}^{2} y_{n - 3} = y_{n - 3}^{3} y_{n - 4}^{2} = y_{n - 4}^{5} y_{n - 5}^{3} = \dots

y_{2} = 1 (1 + \sqrt{2}), y_{3} = {(1 + \sqrt{2})}^{2}, y_{4} = {(1 + \sqrt{2})}^{3}, y_{5} = {(1 + \sqrt{2})}^{5} \dots

1 1 2 3 5 8 13 21 \dots f i b o n a c c i

y_{n} = {(1 + \sqrt{2})}^{f (n)} {\begin{cases} f (n) = f (n - 1) + f (n - 2) \\ f (0) = 0, f (1) = 1 \end{cases}

f (n) = \frac{1}{\sqrt{5}} {{(\frac{1 + \sqrt{5}}{2})}^{n} - {(\frac{1 - \sqrt{5}}{2})}^{n}}

= \frac{ϕ^{n} - {(- ϕ)}^{- n}}{\sqrt{5}}, (ϕ = \frac{1 + \sqrt{5}}{2})

x_{n} = \frac{1}{2} {{(1 + \sqrt{2})}^{f (n)} - {(1 + \sqrt{2})}^{- f (n)}}

+ + + + + + + + + + + +

x_{0} = \frac{1}{2} {{(1 + \sqrt{2})}^{0} - {(1 + \sqrt{2})}^{0}} = 0

x_{1} = \frac{1}{2} {1 + \sqrt{2} - \frac{1}{1 + \sqrt{2}}} = \frac{{(1 + \sqrt{2})}^{2} - 1^{2}}{2 (1 + \sqrt{2})} = 1

x_{2} = x_{1} = 1

x_{3} = \frac{1}{2} {{(1 + \sqrt{2})}^{2} - \frac{1}{{(1 + \sqrt{2})}^{2}}} = \frac{{({(1 + \sqrt{2})}^{2})}^{2} - 1^{2}}{2 {(1 + \sqrt{2})}^{2}}

= \frac{(3 + 2 \sqrt{2} + 1) (3 + 2 \sqrt{2} - 1)}{2 {(1 + \sqrt{2})}^{2}}

= \frac{(2 \sqrt{2} (1 + \sqrt{2})) (2 (1 + \sqrt{2}))}{2 {(1 + \sqrt{2})}^{2}} = 2 \sqrt{2}

\dots

Commented by sou1618 last updated on 03/Sep/16

i f y o u n e e d f i b o n a c c i = f (n)

a_{0} = 0, a_{1} = 1

a_{n} = a_{n - 1} + a_{n - 2} (n ⩾ 2)

a_{n} - {p a}_{n - 1} = q (a_{n - 1} - {p a}_{n - 2})

a_{n} = (p + q) a_{n - 1} - {p q a}_{n - 2}

{\begin{cases} p + q = 1 \\ p q = - 1 \end{cases}

p^{2} - p + p = 0

p = \frac{1 \pm \sqrt{5}}{2}

(p, q) = (\frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2})

{\begin{cases} a_{n} - {p a}_{n - 1} = q (a_{n - 1} - {p a}_{n - 2}) \\ a_{n} - {q a}_{n - 1} = p (a_{n - 1} - {q a}_{n - 2}) \end{cases}

w h e n n = 2

{\begin{cases} a_{1} - {p a}_{0} = 1 \\ a_{1} - {q a}_{0} = 1 \end{cases}

s o

{\begin{cases} a_{n} - {p a}_{n - 1} = q^{n - 1} \cdot \cdot \cdot (1) \\ a_{n} - {q a}_{n - 1} = p^{n - 1} \cdot \cdot \cdot (2) \end{cases}

(2) - (1) \Rightarrow

(p - q) a_{n - 1} = p^{n - 1} - q^{n - 1}

a_{n} = \frac{1}{p - q} (p^{n} - q^{n})

a_{n} = \frac{1}{\sqrt{5}} {{(\frac{1 + \sqrt{5}}{2})}^{n} - {(\frac{1 - \sqrt{5}}{2})}^{n}}

f (n) = a_{n}

Commented by Yozzia last updated on 03/Sep/16

N i c e l y! W h a t i f y o u l e t x_{n} = s i n h {u (n)} i n i t i a l l y ?

Commented by sou1618 last updated on 03/Sep/16

{i t}^{'} s n i c e i d e a!!

x_{n} = s i n h (u_{n})

s i n h (u_{n + 1}) = s i n h (u_{n}) c o s h (u_{n - 1}) + s i n h (u_{n - 1}) c o s h (u_{n})

s i n h (u_{n + 1}) = s i n h (u_{n} + u_{n - 1})

u_{n + 1} = u_{n} + u_{n - 1}

u_{0} = 0

u_{1} = l n (1 + \sqrt{2})

s o

u_{n} = l n (1 + \sqrt{2}) \times f (n)

x_{n} = s i n h {l n (1 + \sqrt{2}) f (n)}

= \frac{1}{2} {{(1 + \sqrt{2})}^{f (n)} - {(1 + \sqrt{2})}^{- f (n)}}

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