If a page is torn from the middle of a book, then the sum of the remaining pages is 718797 so what is the number of torn pages?


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Question Number 115396 by Sudip last updated on 25/Sep/20

$If a page is torn from the middle of a book, then$ $the sum of the remaining pages is 718797 so$ $what is the number of torn pages ?$
	Answered by PRITHWISH SEN 2 last updated on 25/Sep/20

	$901 & 902$
	Answered by TANMAY PANACEA last updated on 25/Sep/20

	$S = \frac{n}{2} [2 \times 1 + (n - 1) 1]$ $S = 718797 + x + x + 1 = \frac{n^{2} + n}{2}$ $n^{2} + n = 2 (2 x + 718798)$ $w a i t . . .$
	Answered by mr W last updated on 26/Sep/20

	$s a y t h e b o o k h a s t o t a l l y n p a g e s .$ $o n e s h e e t w i t h p a g e n u m b e r m a n d$ $m + 1 i s t o r n .$ $\frac{n (n + 1)}{2} - (2 m + 1) = 718797$ $n^{2} + n - 4 (m + 359399) = 0$ $n = \frac{- 1 + \sqrt{1 + 16 (m + 281 \times 1279)}}{2}$ $n > \frac{- 1 + \sqrt{1 + 16 \times 281 \times 1279}}{2} > 1198$ $Δ = 1 + 16 (m + 281 \times 1279) = {(2 k + 1)}^{2}$ $\Rightarrow n = k$ $4 (m + 281 \times 1279) = k (k + 1)$ $\Rightarrow k = 4 h$ $\Rightarrow m + 281 \times 1279 = h (4 h + 1)$ $m < n = k = 4 h$ $m = h (4 h + 1) - 281 \times 1279 < 4 h$ $4 h^{2} - 3 h - 281 \times 1279 < 0$ $\Rightarrow h < \frac{3 + \sqrt{9 + 16 \times 281 \times 1279}}{8} \approx 300.1$ $\Rightarrow h ⩽ 300$ $4 h = n > 1198$ $\Rightarrow h > \frac{1198}{4} = 299.5$ $\Rightarrow h ⩾ 300$ $w e s e e t h e o n l y s o l u t i o n i s h = 300 :$ $n = 4 h = 1200$ $m = h (4 h + 1) - 281 \times 1279 = 901$ $\Rightarrow t h e b o o k h a s 1200 p a g e s,$ $p a g e 901 a n d 902 a r e t o r n .$
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