U_n = ((((−4)^(n+1) −1)/(1−(−4)^n )))U_(n−1) with U_0 =1 find U


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Question Number 173132 by alcohol last updated on 07/Jul/22

$U_{n} = (\frac{{(- 4)}^{n + 1} - 1}{1 - {(- 4)}^{n}}) U_{n - 1} w i t h U_{0} = 1$ $f i n d U_{n} i n t e r m s o f n$
Commented by kaivan.ahmadi last updated on 07/Jul/22

$w e h a v e \frac{u_{n}}{u_{n - 1}} = \frac{{(- 4)}^{n + 1} - 1}{1 - {(- 4)}^{n}}$ $o n t h e o t h e r h a n d$ $u_{1} = (\frac{16 - 1}{1 + 4}) 1 = 3$ $u_{2} = (\frac{- 64 - 1}{1 - 16}) 3 = 13$ $u_{3} = (\frac{256 - 1}{1 + 64}) 13 = 51$ $u_{4} = (\frac{- 1024 - 1}{1 - 256}) 51 = 205$ $\begin{array}{ccc} n 1 2 3 4 \\ 4 \times 1 - 1 4 \times 3 + 1 4 \times 13 - 1 4 \times 51 + 1 \\ u_{n} 3 13 51 205 \end{array}$ $s o u_{n} = 4 u_{n - 1} + {(- 1)}^{n}$ $\Rightarrow \frac{u_{n}}{u_{n - 1}} = 4 + \frac{{(- 1)}^{n}}{u_{n - 1}} \Rightarrow \frac{{(- 4)}^{n + 1} - 1}{1 - {(- 4)}^{n}} = 4 + \frac{{(- 1)}^{n}}{u_{n - 1}}$ $\Rightarrow \frac{{(- 4)}^{n + 1} - 1}{1 - {(- 4)}^{n}} - 4 = \frac{{(- 1)}^{n}}{u_{n - 1}} \Rightarrow$ $\frac{{(- 4)}^{n + 1} - 1 - 4 - {(- 4)}^{n + 1}}{1 - {(- 4)}^{n}} = \frac{{(- 1)}^{n}}{u_{n - 1}}$ $\Rightarrow \frac{- 5}{1 - {(- 4)}^{n}} = \frac{{(- 1)}^{n}}{u_{n - 1}} \Rightarrow u_{n - 1} = \frac{{(- 1)}^{n} [{(- 4)}^{n} - 1]}{5}$ $r e p l a c e n w i t h n + 1$ $u_{n} = \frac{{(- 1)}^{n + 1} [{(- 4)}^{n + 1} - 1]}{5}$
Commented by Tawa11 last updated on 07/Jul/22

$Great sir .$
Commented by kaivan.ahmadi last updated on 07/Jul/22

$t h a n k y o u s o m u c h$
Commented by mr W last updated on 06/Jan/23

$t h a n k s s i r!$ $y o u r a n s w e r i s c o r r e c t, b u t s o l u t i o n$ $i s n o t m a t h e m a t i c a l l y s t r i c t .$ $y o u c a l c u l a t e d s o m e v a l u e s f o r a_{n} :$ $a_{1} = 3, a_{2} = 13, a_{3} = 51, a_{4} = 205,$ $a n d t h e n s a y u_{n} = 4 u_{n - 1} + {(- 1)}^{n} .$ $t h i s m e t h o d i s n o t m a t h e m a t i c a l l y$ $s t r i c t . {i t}^{'} s s o m e t i m e s a m a t t e r o f$ $l u c k t h a t w e ‘ ‘ {g u e s s}^{″} t h e r i g h t$ $a n s w e r . a f a m o u s e x a m p l e :$ $a_{1} = 1, a_{2} = 2, a_{3} = 4, a_{4} = 8, a_{5} = 16, . . .$ ${i t}^{'} s t h e l a r g e s t n u m b e r o f p o r t i o n s$ $o f a c i r c l e w h i c h c a n b e d i v i d e d b y n$ $p o i n t s o n t h e c i r c l e . c a n w e s a y f r o m$ $t h e s e v a l u e s t h a t a_{n} = 2^{n - 1} ?$ $u n f o r t u n a t e l y n o! e v e n i t d e l i v e r s t h e$ $c o r r e c t v a l u e s f o r t h e f i r s t 5 c a s e s .$ $t h e c o r r e c t f o r m u l a i s a_{n} = C_{4}^{n} + C_{2}^{n} + 1 .$ $t h i s e x a m p l e s h o w s h o w d a n g e r o u s$ $i t i s w h e n w e j u s t l o o k a t s o m e$ $f i r s t v a l u e s a n d t h e n g u e s s a$ $g e n e r a l f o r m u l a f o r t h e n^{t h} t e r m .$ $i f y o u a r e i n t e r e s t e d, y o u m a y r e a d$ $m o r e a b o u t t h i s e x a m p l e i n Q 180457 .$
Commented by mr W last updated on 06/Jan/23

$a m a t h e m a t i c a l l y s t r i c t s o l u t i o n$ $f o r t h i s q u e s t i o n s e e Q 184401 .$
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