Pages Of A Book

Sine, sum of two consecutive numbers can not be the even number.
So, 90 and 92 can't be correct.
99,95 & 97 are sum of two consecutive numbers (49 and 50),(47 and 48) and (48 and 49) respectively.

According to the pattern of the page number of book (49 and 50) and (47 and 48) will be on two sides of a same paper but (48 and 49) faces each other.

So 97 is the only possible sum of two facing pages.

Those are page numbers 48 and 49

Those must be page number 48 and 49

Set up an arithmetic sequence for the sum of 2 pages which face each other.

First term is 1 (Page one is on its own)

2nd term: 2 + 3 = 5

3rd term: 4 + 5 = 9

We have the sequence 1,5,9,13,... etc. Therefore, we can say that the nth term will have a sum of 4n-3. All we now need to do is look at the choices, add 3 to each one and see which one is divisible by 4.

Well, the two pages will be consecutive. So we can represent the sum as x+x+1 simplified to 2x+1. Also, since pages 2&3 and 4&5 face each other, we can logically assume that "x" will be even. Therefore, we just solve each potential answer, until that happens. e.g. 2x+1=97, 2x=96, x= $\frac{96}{2}$ =48 48 is even, therefore it is a possible solution.

The facing pages have numbers $2n$ and $2n+1$ for some positive integer $n$ . Their sum is therefore $4n+1$ .

The solution must therefore be one more than a multiple of four. Only $\color{#D61F06}{97}$ satisfies that condition.

sum of two consecutive number must be odd number, so 90 and 92 are rejected.

Observe the mean of two facing page number, the integer part must be an even number.

i.e., (2+3)/2 = 2.5, the integer part is 2, an even number

(4+5)/2 = 4.5, the integer part is 4, an even number

Making use of this pattern, we can figure out the correct answer by dividing each option by 2:

97/2=48.5, the integer part is even, so 97 is the answer.

95/2=47.5, the integer part is odd, rejected.

99/2=49.5, the integer part is odd, rejected.

sum of two consecutive numbers can not be an even number. So, 90 and 92 can't be correct. Furthemore, if 95 was a solution ,99 would be another solution, so the only possibility is 97 if the problem is correct.

One page will have the form 2k. The other page will have the form 2k+1 where k is an integer.

2k is an even integer 2k+1 is an odd integer.

The sum of the two pages facing each other is always odd. Therefore 90 and 92 can not be the sums.

The possible sums are 95,97,99.

47+48 = 95 48+49 = 97 49+50 = 99

The even page number always comes first in pages that are facing each other in this book. Therefore the correct answer is 48+49 = 97.