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-

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 \dots

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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