Question

From a book, a number of consecutive pages more than 1 are missing. The sum of page numbers of these pages is 9808.What are the page numbers of missing pages?

Answer 1

$$\begin{gathered} \text{Let total number of missing sheets=n} \\ \text{Let start with page x and ending with 2n consecutive page number}\left(\text{because, each sheet has two page}\right)\text{. i.e.,} \\ x,\left(x+1\right),\left(x+2\right),.....x+\left(2n-1\right) \\ \text{According to question,} \\ x+\left(x+1\right)+\left(x+2\right),.....x+\left(2n-1\right)=9808 \\ \Rightarrow \frac{2n}{2}\left(2x+\left(2n-1\right)1\right)=9808 \\ \Rightarrow n\left(2x+2n-1\right)=9808 \\ \Rightarrow n\left(2x+2n-1\right)=2^4\times 613 \\ \text{Since 2x+2n-1 is odd}\left(\text{even+even-odd=odd}\right)\text{therefore,n is even} \\ \Rightarrow n=2^4=16 \\ \text{And,} \\ 2x+2\times 16-1=613 \\ \Rightarrow 2x=613-32+1 \\ \Rightarrow 2x=582 \\ \Rightarrow x=291 \\ \text{Now, the number of pages missing is twice the number of sheets missing. So,} \\ 2n=2\times 16=32 \\ \text{Thus, 32 consecutive pages are missing. i.e., 291 to 322.} \end{gathered}$$