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Question Number 27300 by mondodotto@gmail.com last updated on 04/Jan/18

Commented by mondodotto@gmail.com last updated on 04/Jan/18

$any one to help this please$
	Answered by bsayani309@gmail.com last updated on 05/Jan/18

	$i f w e t h i n k t h a t 3 = x t h e n t h e s e q u e n c e m a y b e$ $\frac{x}{x + 2} . \frac{2 (x + 2)}{2 (x + 2) - 2} . \frac{2 [2 (x + 2) - 2]}{2 [2 (x + 2) - 2] + 2}$ $l i k e t h i s t h e 5 t h t e r m b e c o m s \frac{68}{70} a n d t h e 6 t h t e r m i s \frac{140}{138}$
	Answered by prakash jain last updated on 05/Jan/18

	$t_{n} = \frac{a_{n}}{b_{n}} (n = 0, 1, 2, 3, . . .)$ $a_{0} = 3$ $b_{0} = 5$ $b_{1} = 8$ $a_{n} = 2 b_{n - 1} \Rightarrow a_{n - 1} = 2 b_{n - 2}$ $b_{n} = a_{n - 1} + b_{n - 1}$ $b_{n} = 2 b_{n - 2} + b_{n - 1}$ $b_{n} - b_{n - 1} - 2 b_{n - 2} = 0$ $x^{2} - x - 2 = 0$ $(x + 1) (x - 2) = 0$ $b_{n} = c_{1} {(- 1)}^{n} + c_{2} 2^{n}$ $b_{0} = 5 \Rightarrow c_{1} + c_{2} = 5$ $b_{1} = 8 \Rightarrow - c_{1} + 2 c_{2} = 8$ $b_{n} = \frac{2}{3} {(- 1)}^{n} + \frac{13}{3} 2^{n} = \frac{13 \cdot 2^{n} + {(- 1)}^{n} \cdot 2}{3} (n ⩾ 0)$ $b_{n} = \frac{13 \cdot 2^{n} + {(- 1)}^{n} \cdot 2}{3} (n ⩾ 0)$ $a_{n} = 2 b_{n - 1} (n ⩾ 1)$ $a_{0} = 3$ $c h e c k$ $b_{0} = \frac{13 + 2}{3} = 5, a_{1} = 2 b_{0} = 10$ $b_{1} = \frac{13 \cdot 2 - 2}{3} = 8, a_{2} = 16$ $b_{2} = \frac{13 \cdot 4 + 2}{3} = 18, a_{3} = 36$ $b_{3} = \frac{13 \cdot 8 - 2}{3} = 34, a_{4} = 68$ $b_{4} = \frac{13 \times 16 + 2}{2} = 70, a_{5} = 140$
	Commented by mondodotto@gmail.com last updated on 05/Jan/18

	$thanx a lot really apreciate this$
	Commented by Rasheed.Sindhi last updated on 05/Jan/18

	$e^{X} c e l e n t S i r!$
	Answered by shiv15031973@gmail.com last updated on 05/Jan/18
	$as every denominator double is next no. numerator ∵34+34=68 and every fraction is having difference at up down of 2 in sequence so next denominator will be70 ans=((68)/(70)) and((140)/(138)) .$
	$a s e v e r y d e n o m i n a t o r d o u b l e i s$ $n e x t n o . n u m e r a t o r ∵ 34 + 34 = 68$ $a n d e v e r y f r a c t i o n i s h a v i n g d i f f e r e n c e$ $a t u p d o w n o f 2 i n s e q u e n c e$ $s o n e x t d e n o m i n a t o r w i l l b e 70$ $a n s = \frac{68}{70} a n d \frac{140}{138} .$
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