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Question Number 81308 by naka3546 last updated on 11/Feb/20

Commented by mr W last updated on 11/Feb/20

$2104 ?$
Commented by naka3546 last updated on 11/Feb/20

$p l e a s e, s h o w y o u r w o r k i n g, s i r .$
	Answered by mr W last updated on 12/Feb/20

	$g i v e n i s t h e r e c u r s i v e r e l a t i o n$ $x_{n + 13} = x_{n + 4} + 2 x_{n}$ $w e s e e x_{n} = {A q}^{n} f u l f i l l s t h i s c o n d i t i o n .$ ${A q}^{n + 13} = {A q}^{n + 4} + 2 {A q}^{n}$ $\Rightarrow q^{13} = q^{4} + 2$ $i t h a s 13 r o o t s, s a y q_{r} (r = 1, 2, . . ., 13) .$ $t h e g e n e r a l f o r m u l a f o r t e r m x_{n} i s t h e n$ $x_{n} = \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{n}$ $w i t h q_{r}^{13} = q_{r}^{4} + 2 (r = 1, 2, 3, . . ., 13)$ $x_{0} = \underset{r = 1}{\sum^{13}} A_{r} = 1 (x_{13} = x_{4} + 2 x_{0} \Rightarrow x_{0} = 1)$ $x_{m} = \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{m} = 0 (m = 1, 2, 3, . . ., 12)$ $x_{13} = \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{13} = 2$ $x_{143} = \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{143}$ $q_{r}^{143} = {(q_{r}^{13})}^{11} = {(q_{r}^{4} + 2)}^{11} = \underset{k = 0}{\sum^{11}} C_{k}^{11} q_{r}^{4 k} 2^{11 - k}$ $x_{143} = \underset{r = 1}{\sum^{13}} A_{r} (\underset{k = 0}{\sum^{11}} C_{k}^{11} q_{r}^{4 k} 2^{11 - k})$ $x_{143} = \underset{k = 0}{\sum^{11}} C_{k}^{11} 2^{11 - k} (\underset{r = 1}{\sum^{13}} A_{r} q_{r}^{4 k})$ $k = 0 \Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{4 k} = 1$ $k = 1 . . 3 \Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{4 k} = 0$ $\Rightarrow x_{134} = 2^{11} + \underset{k = 4}{\sum^{11}} C_{k}^{11} 2^{11 - k} (\underset{r = 1}{\sum^{13}} A_{r} q_{r}^{4 k})$ $k = 4 . . 11 :$ $q_{r}^{16} = q_{r}^{13} q_{r}^{3} = (q_{r}^{4} + 2) q_{r}^{3} = q_{r}^{7} + 2 q_{r}^{3}$ $\Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{16} = \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{7} + 2 \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{3} = 0$ $q_{r}^{20} = q_{r}^{13} q_{r}^{7} = (q_{r}^{4} + 2) q_{r}^{7} = q_{r}^{11} + 2 q_{r}^{7}$ $\Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{20} = \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{11} + 2 \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{7} = 0$ $q_{r}^{24} = q_{r}^{13} q_{r}^{11} = (q_{r}^{4} + 2) q_{r}^{11} = (q_{r}^{4} + 2) q_{r}^{2} + 2 q_{r}^{11}$ $\Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{24} = 0$ $q_{r}^{28} = {(q_{r}^{13})}^{2} q_{r}^{2} = {(q_{r}^{4} + 2)}^{2} q_{r}^{2} = q_{r}^{10} + 4 q_{r}^{6} + 4 q_{r}^{2}$ $\Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{28} = 0$ $q_{r}^{32} = {(q_{r}^{13})}^{2} q_{r}^{6} = {(q_{r}^{4} + 2)}^{2} q_{r}^{6} = q_{r}^{14} + 4 q_{r}^{10} + 4 q_{r}^{6}$ $= (q_{r}^{4} + 2) q_{r} + 4 q_{r}^{10} + 4 q_{r}^{6}$ $\Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{32} = 0$ $q_{r}^{36} = {(q_{r}^{13})}^{2} q_{r}^{10} = {(q_{r}^{4} + 2)}^{2} q_{r}^{10} = (q_{r}^{8} + 4 q_{r}^{4} + 4) q_{r}^{10}$ $= q_{r}^{18} + 4 q_{r}^{14} + 4 q_{r}^{10}$ $= (q_{r}^{4} + 2) q_{r}^{5} + 4 (q_{r}^{4} + 2) q_{r} + 4 q_{r}^{10}$ $\Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{36} = 0$ $q_{r}^{40} = {(q_{r}^{13})}^{3} q_{r} = {(q_{r}^{4} + 2)}^{3} q_{r} = (q_{r}^{12} + 6 q_{r}^{8} + 12 q_{r}^{4} + 8) q_{r}$ $= q_{r}^{13} + 6 q_{r}^{9} + 12 q_{r}^{5} + 8 q_{r}$ $\Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{40} = \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{13} = 2$ $q_{r}^{44} = {(q_{r}^{13})}^{3} q_{r}^{5} = {(q_{r}^{4} + 2)}^{3} q_{r}^{5} = (q_{r}^{12} + 6 q_{r}^{8} + 12 q_{r}^{4} + 8) q_{r}^{5}$ $= q_{r}^{17} + 6 q_{r}^{13} + 12 q_{r}^{9} + 8 q_{r}^{5}$ $= (q_{r}^{4} + 2) q_{r}^{4} + 6 q_{r}^{13} + 12 q_{r}^{9} + 8 q_{r}^{5}$ $\Rightarrow \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{44} = 6 \underset{r = 1}{\sum^{13}} A_{r} q_{r}^{13} = 6 \times 2 = 12$ $x_{143} = 2^{11} + C_{10}^{11} 2^{1} (\underset{r = 1}{\sum^{13}} A_{r} q_{r}^{40}) + C_{11}^{11} 2^{0} (\underset{r = 1}{\sum^{13}} A_{r} q_{r}^{44})$ $x_{143} = 2^{11} + 11 \times 2 \times 2 + 1 \times 2^{0} \times 12$ $\Rightarrow x_{143} = 2^{11} + 44 + 12 = 2104$ $i n t h i s w a y w e c a n c a l c u l a t e e v e r y$ $t e r m o f t h e s e q u e n c e e v e n t h o u g h$ $w e {d o n}^{'} t k n o w t h e v a l u e s o f A_{r} a n d$ $q_{r} (r = 1, 2, . . ., 13) .$
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