let-p-q-be-reals-such-that-p-gt-q-gt-0-define-the-sequence-x-n-where-x-1-p-q-and-x-n-x-1-pq-x-n-1-for-n-2-for-all-n-then-x-n-

Question Number 210868 by universe last updated on 20/Aug/24

let p, q be reals such that p > q > 0 define

the sequence {x_{n}} where x_{1} = p + q and

x_{n} = x_{1} - \frac{pq}{x_{n - 1}} for n ⩾ 2 for all n then x_{n} = ? ?

Commented by Ghisom last updated on 20/Aug/24

after calculating a few x_{k} I get

x_{n} = \frac{p^{n + 1} - q^{n + 1}}{p^{n} - q^{n}}

Commented by universe last updated on 20/Aug/24

c a n u s e n d y o u r s o l u t i o n

Commented by mr W last updated on 21/Aug/24

i f o u n d m y o w n m e t h o d f o r s o l v i n g

x_{n} = a + \frac{b}{x_{n - 1}} a n d g o t f o r t h i s q u e s t i o n

a l s o x_{n} = \frac{p^{n + 1} - q^{n + 1}}{p^{n} - q^{n}} .

Commented by universe last updated on 21/Aug/24

s i r p l e a s e s e n d y o u r s o l u t i o n

Commented by mm1342 last updated on 21/Aug/24

b y i n d u c t i o n

f o r n = 1 \Rightarrow x_{1} = \frac{p^{2} - q^{2}}{p - q} = p + q ✓

n = k \to x_{k} = \frac{p^{k + 1} - q^{k}}{p^{k} - q^{k}}

n = k + 1 \to x_{k + 1} = p + q - \frac{p q}{x_{k}}

= p + q - \frac{p q (p^{k} - q^{k})}{p^{k + 1} - q^{k + 1}}

= \frac{p^{k + 2} - {p q}^{k + 1} + {q p}^{k + 1} - q^{k + 2} - p^{k + 1} q + {p q}^{k + 1}}{p^{k + 1} - q^{k + 1}}

= \frac{p^{k + 2} - q^{k + 2}}{p^{k + 1} - q^{k + 1}} ✓

Answered by mr W last updated on 21/Aug/24

t h i s i s m y m e t h o d :

x_{n} = p + q - \frac{p q}{x_{n - 1}} = k - \frac{h}{x_{n - 1}}

l e t x_{n} = \frac{a_{n}}{b_{n}}

\frac{a_{n}}{b_{n}} = k - \frac{{h b}_{n - 1}}{a_{n - 1}} = \frac{{k a}_{n - 1} - {h b}_{n - 1}}{a_{n - 1}}

b_{n} = a_{n - 1}

a_{n} = {k a}_{n - 1} - {h b}_{n - 1} = {k a}_{n - 1} - {h a}_{n - 2}

\Rightarrow a_{n} - {k a}_{n - 1} + {h a}_{n - 2} = 0

t h i s i s a r e c u r r e n c e r e l a t i o n .

i t s c o r r e s p o n d i n g c h a r a c t e r i s t i c

e q u a t i o n i s

r^{2} - k r + h = 0 \Rightarrow r^{2} - (p + q) r + p q = 0

\Rightarrow r = p, q

g e n e r a l s o l u t i o n f o r a_{n} i s

a_{n} = {A p}^{n} + {B q}^{n} w i t h c o n s t a n t s A, B

\Rightarrow b_{n} = a_{n - 1} = {A p}^{n - 1} + {B q}^{n - 1}

g e n e r a l s o l u t i o n f o r x_{n} i s t h e n

x_{n} = \frac{a_{n}}{b_{n}} = \frac{{A p}^{n} + {B q}^{n}}{{A p}^{n - 1} + {B q}^{n - 1}} = \frac{{C p}^{n} + q^{n}}{{C p}^{n - 1} + q^{n - 1}}

g i v e n : x_{1} = \frac{C p + q}{C + 1} = p + q

\Rightarrow p + C q = 0 \Rightarrow C = - \frac{p}{q}

\Rightarrow x_{n} = \frac{(- \frac{p}{q}) p^{n} + q^{n}}{(- \frac{p}{q}) p^{n - 1} + q^{n - 1}}

o r x_{n} = \frac{p^{n + 1} - q^{n + 1}}{p^{n} - q^{n}} ✓

Commented by mr W last updated on 21/Aug/24

t h i s i s a g e n e r a l m e t h o d w h i c h c a n

s o l v e r e c u r r e n c e r e l a t i o n s l i k e

x_{n} = a + \frac{b}{x_{n - 1}} .

Commented by universe last updated on 21/Aug/24

t h a n k y o u s o m u c h s i r